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For more information, please see full course syllabus of Probability
For more information, please see full course syllabus of Probability
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Post by Thuy Nguyen on November 4, 2016
Hi Dr. Murray, thank you for your lecture, you explain so well. On the integral of finding E(Y1), your 1 looks like a 2, and I was confused until the end where you circled the number and said it was a, "one".
Post by Carl Scaglione on June 13, 2014
Professor Murray, On page 5, referring to the last equation, the summation terms are i>j. Why not show i not equal to j?
Covariance, Correlation & Linear FunctionsDownload Quick Notes
Covariance, Correlation & Linear Functions
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Covariance is a Measure of Dependence
- Dependence Doesn't Necessarily Mean that the Variables Do the Same Thing
- If Variables Move Together
- If Variables Move Against Each Other
- Both Cases Show Dependence!
- Intro 0:00
- Definition and Formulas for Covariance 0:38
- Definition of Covariance
- Formulas to Calculate Covariance
- Intuition for Covariance 3:54
- Covariance is a Measure of Dependence
- Dependence Doesn't Necessarily Mean that the Variables Do the Same Thing
- If Variables Move Together
- If Variables Move Against Each Other
- Both Cases Show Dependence!
- Independence Theorem 8:10
- Independence Theorem
- The Converse is Not True
- Correlation Coefficient 9:33
- Correlation Coefficient
- Linear Functions of Random Variables 11:57
- Linear Functions of Random Variables: Expected Value
- Linear Functions of Random Variables: Variance
- Linear Functions of Random Variables, Cont. 14:30
- Linear Functions of Random Variables: Covariance
- Example I: Calculate E (Y₁), E (Y₂), and E (Y₁Y₂) 15:31
- Example II: Are Y₁ and Y₂ Independent? 29:16
- Example III: Calculate V (U₁) and V (U₂) 36:14
- Example IV: Calculate the Covariance Correlation Coefficient 42:12
- Example V: Find the Mean and Variance of the Average 52:19
Introduction to Probability Online Course
Transcription: Covariance, Correlation & Linear Functions
Hi, welcome back to the probability lectures here on www.educator.com.0000
We are working through a chapter on Bivariate distribution functions and density functions,0004
which means that there are two variables, there is a Y1 and Y2.0010
In this section, we are also going to have sometimes more than two variables that might be N variables.0016
We got a big section to cover today, it is going to cover covariance and correlation coefficient,0022
and linear functions of random variables.0029
I will be your guide today, my name is Will Murray, let us jump right on in.0032
The main idea of this section is covariance, the correlation coefficient is not something that is quite as important.0040
Let me jump right on in with the covariance.0046
The definition of covariance is not necessarily very enlightening.0048
Let me go ahead and show you the definition, but then I'm going to skip quickly to some formulas0054
that are probably more useful in dealing with covariance.0059
In the next slide, I will try to give you an intuition for what covariance means.0063
The definition of the covariance is that, you will have to start with two random variables.0068
You always have a Y1 and Y2.0075
You always talk about the covariance of two random variables at once.0078
By definition, it means the expected value of Y1 - μ1 and what × Y2 - μ2.0082
Here, μ1 and μ2 are the means or the expected values of Y1 and Y2.0091
I think that, that definition does not offer a lot of intuitive light onto what covariance means.0098
I will talk about the intuition, maybe on the next slide.0105
In the meantime, I will give you some useful formulas for calculating covariance,0108
because that definition is also not very useful for calculating covariance.0113
Often, the easiest way to calculate covariance is to use this formula right here, where you calculate the expected value of Y1 × Y2.0117
And then, you subtract off expected value of Y1 × the expected value of Y2.0128
That is usually the easiest way to calculate it.0134
By the way, for each one of these, you are going to have to calculate the expected value of a function of random variables.0136
We learned how to do that, in the previous lecture.0144
If you are not sure how you would calculate the expected value for example of Y1 × Y2,0147
what you want to do is watch the previous lecture here on the probability series on www.educator.com.0153
I went through some examples where we practiced calculating things like that.0161
You can see how you would calculate that.0166
A useful point here is that, if you ever have to calculate the covariance of Y1 with itself,0169
it is exactly equal to the variance of Y1.0176
If you have to calculate the covariance of any single variable with itself, it is just the same as the variance.0180
The way covariance behaves under scaling is that, if you multiply these variables by constants0186
then those constants just pop out, and you just get C² coming out to the outside.0193
That is a very useful if you have to deal with linear functions which is something0198
we are going to talk about later on, in this lecture.0203
That is the definition, which I do not really recommend that you use that often.0208
The definition for me is not useful, when calculating covariance.0212
These formulas are much more useful, specially this first one, it is the one that I used all the time,0216
when I'm calculating covariance.0221
That is definitely worth committing to memory.0223
I have not really told you yet, what covariance means when you are measuring.0227
Let me spend the next slide talking about that.0233
The intuition for covariance is that, it is a measure of dependents between your two variables Y1 and Y2.0236
It measures how closely Y1 and Y2 track each other.0245
There is an easy mistake that students make, when you are first learning about probability,0254
which is to think that dependence,0259
If two variables are dependent that means that they do the same thing.0261
That is not quite what dependence means, what dependence means is that knowing about what one variable does,0265
gives you more information about what the other variable does.0273
It does not necessarily mean that they move together.0278
It just means that one variable gives you some kind of guide, as to what the other variable is doing.0281
The way that behaves, in terms of covariance, is that if the variables do move together,0287
then covariance will be positive.0294
It shows that those variables are positively correlated.0296
If one is big then you expect the other one to be big.0300
If Y1 moves consistently against Y2, which means Y1 is big and the other one is small.0305
Or if the first one is small the other one is big.0312
They move like this, they move against each other, that is still dependence.0314
That would be reflected in the covariance, you will get a negative value for the covariance.0322
It means these variables are negatively correlated against each other.0327
Let me emphasize here that both of these are still considered to be dependence, to be examples of dependence.0331
Both of these show dependence.0341
And that is sometimes a little confusing for students, when you are first learning about probability.0349
Both of these are examples of dependence.0354
You think, wait a second, if two variables are moving against each other, are not those independent,0357
that kind of make you think of your intuition as being independent, if they are moving against each other.0363
Not so, that is dependent, if they are moving against each other.0368
The example that I use with my own students is, to imagine that you are a parent and0373
you have 2 twin children, may be 2 twin boys.0378
One of your children is just very well behaved, the boy does everything that you tell him to.0383
That is dependence because you can sort of control what that boy does,0390
by telling him to do something and he does it.0394
Imagine that your other twin boy is very mischievous.0397
He always does the opposite of what you tell him to do.0401
Whatever you tell him to do, he does the opposite.0404
If you tell him to go to bed, then he runs around and plays.0407
If you tell him to runaround and play, then he goes to bed.0411
You might think, that is a very independent child but that is not an independent child0416
because you can still control that child by using reverse psychology.0422
If you want him to go to bed, tell him to run around and play, and he will go to bed.0426
If you want him to run around and play, tell him to go to bed and he will run around and play.0431
You can still control that child because that child is still responding to your commands,0436
he is just responding in the opposite fashion.0442
You can still control that child, just by using reverse psychology.0444
That is dependence and that is kind of the situation that you want to think of here,0448
when you got two variables that move against each other.0453
If you had a child that, if you tell him to go to bed, sometimes he goes to bed and sometimes he runs around and plays,0457
that would be an independent child, that would be a child you could not control by any kind of psychology.0463
That is really what you want think of independence.0470
If you cannot control the actions then that would be independence.0472
But, if two variables move together, that is dependence.0477
If two variables move against each other consistently, that is still dependence.0482
Finally, let me show you how independence enters this picture.0492
The theorem is that if Y1 and Y2 are independent then their covariance is always going to be equal to 0.0496
Remember, covariance is the measure of dependence.0504
If they are independent then the covariance is 0.0508
Unfortunately, the converse of that theorem is not true.0512
You would like to say, if the covariance is 0 then the variables are independent.0516
That is not true, you can have covariance of two variables being 0 and that still have some dependence between the variables.0521
That is a rather unfortunate result of the mathematics, there is no way for me to fix that.0533
I will give you an example of that and that is coming up in the problems that we are about to do, in example 2.0540
If you just scroll down, scroll forward in this lecture, you will find an example0549
where the covariance is going to come out to be 0, and the Y1 and Y2 will still be dependent.0556
That is kind of unfortunate, it would be very nice if this theorem worked in both directions.0564
It does work in one direction but it does not work in the other direction.0568
One new concept for this lecture is the correlation coefficient.0575
This is very closely related to covariance.0580
In fact, if you are studying along in your book, they probably be mentioned in the same section with covariance in your book.0583
You start out with random variables, you calculate their covariance.0590
Remember, we learn about that a couple slides ago.0594
You can go back and look at the definition of covariance.0597
What you do is you just divide by their standard deviations.0600
The correlation coefficient is really just a scaled version of the covariance.0604
You just take the covariance and you scale it down, by a couple of constants.0611
The point of the correlation coefficient is that, if you did multiply each one of your variables by a constant0615
then the constant washes out of the definition of the correlation coefficient.0624
You end up with the same correlation coefficient, that you would have in the first place.0630
By the way, this Greek letter is pronounced ρ.0636
This is the Greek letter ρ, ρ of a scale version of the variables comes out to be the same ρ.0639
That is very nice, that means that ρ, the correlation coefficient is independent of scale.0648
It is convenient, if you are taking measurements.0654
It does not matter, if you are measuring in inches, feet, or meters, or whatever.0657
You are still going to get the same correlation coefficient.0660
In particular, the correlation coefficient will always be between -1 and 1.0664
It is an absolute constant that, you can discuss correlation coefficients between different sets of data.0670
You will always know that you are working on a scale between -1 and 1.0678
That is not true with covariance, covariance can be as big or as be negative as you can imagine.0682
But, correlation coefficient is always between -1 and 1, it is a sort of a universal scale.0689
There is going to be an example in this lecture where we will calculate the correlation coefficient.0696
At the end, we will actually translate into a decimal and we will just check if that it is between -1 and 1.0701
If it does come out to be between -1 and 1, then that is a little signal that we have probably done our work right.0706
If it comes out to be bigger than 1, we know that we have done something wrong.0713
The next topic that we need to learn about is linear functions of random variables.0717
Let us start out with a collection of random variables, Y1 through YN.0723
We have means or expected values, remember expected value and mean are the exact same thing.0727
Means are μ1 to μn and the variances are σ1² through σ N².0735
What we want do is build up this linear combination, this linear function A1 Y1 up to AN YN.0743
We are building a linear function out of the Yi.0751
We want to find the expected value of that construction.0756
It turns out to be just exactly what you would think and hope it would be,0759
which is just A1 × expected value of Y1 up to AN × the expected value of YN.0764
That is just because expectation is linear, it works very well and gives you what you would expect.0772
The variance is not so nice, it is a little bit trickier.0780
The variance of A1 Y1 up to AN YN, first of all, these coefficients get².0784
You have A1² and AN².0792
And then, the σ 1², remember that is the variance of Y1 up to σ N² is the variance of YN.0796
That is not all, there is another term on this formula which is that,0809
you have to look at all the covariances of all the variables with each other.0813
You look at all the covariances of the i and j, and for each pair, if you have Y1 and Y3 or Y2 and Y5,0819
for each one of those pairs, you take the coefficients of each one, the Ai and Aj.0830
You add them up and you multiply all these by 2.0836
The reason you are multiplying by 2 is because you are doing Y1 with Y3.0839
And then, later on you will be doing Y3 with Y1.0844
That is why we get that factor of 2 in there, it is because you get each pair in each order.0848
That is what you would get for the variance of a linear combination of random variables.0857
We will study some examples of that, so you got a chance to practice this.0864
There is one more formula which is, when you want to calculate the covariance of two linear combinations.0868
The covariance of A1 Y1 up to AN YN and B1 X1 up to BM XM, the covariance of those two things together.0876
It actually behaves very nicely, you just take the covariance of all the individual pairs.0890
You can factor out the coefficients and then you just add up that sum.0896
All the pairs, YI × XJ or covariance of YI with XJ, and then you just put on the coefficients Ai and B sub J.0901
The covariance behaves really quite nicely with respect to linear combinations.0914
That is a lot of background material, I hope that I have not lost you yet.0920
I want to jump in and we will solve some examples, and we will see how all these formulas play out in practice.0925
In example 1, we have got a joint density function.0933
This terminology might be a little unfamiliar to people who are just joining me.0938
That colon means defined to be.0942
We are defining the joint density to be and that = means always equal to.0949
It is like equal but it is sort of saying that, no matter what Y1 and Y2 are, this is always equal to 1.0959
Let us see, this is over the triangle with corners at -1, 0 and 0, 1 and 1,0.0966
I will go ahead and graph that because we are going to end up calculating some double integrals here.0975
Let me use the formulas that we learned in the previous lecture, on expected values of functions of random variables.0981
If you did not watch that lecture, you really want to go back and watch that lecture before this example will make sense.0989
There is the -1,0 and there is 0,1 and there is 1,0.0999
The region we are talking about here, let me go ahead and put my scales on here.1007
This is Y1 and this is Y2, there is -1, there is 1, there is 1, and 0.1014
This region that we are talking about is this triangular region, that is sort of a triangle here.1023
Since, I'm going to be using some double integrals to calculate these expected values,1031
I want to describe this region.1036
I think the best way to describe it, is by listing Y2 first and then listing Y1 as varying between these two lines.1039
Otherwise, I would have to chop this up into two separate pieces.1049
It is really more work than what I want to do.1052
Let me try to find the equation of those lines.1055
The sign back here is Y2 = Y1 + 1.1057
That is just following slope intercept form, the slope is 1 and Y intercept is 1.1063
It is like Y = X + 1.1070
If I solve for that in terms of Y1, that is Y1 = Y2 -1.1072
This line right here is, the slope Y2 is slope -1 - Y1 + 1.1079
If I solve for Y1, I get Y1 would be 1 - Y2, that is that line.1089
If I want to describe that region, I can describe it in terms of Y2 first.1097
Y2 goes from 0 to 1 and Y1 goes from that left hand diagonal line Y2 -1, to the right hand diagonal line which is 1 - Y2.1101
I got a description of the region, I need to set up some double integrals.1121
Let me set up a double integral for expected value of Y1.1125
All these double integrals will have the same bound, that is one small consolation.1130
To get the value of Y1, will be the integral as Y2 goes from 0 to 1.1135
Y1 goes from Y2 -1 to 1 - Y2, just following those limits there.1142
I'm calculating the expected value of Y1.1155
I will put Y1 here and I want to put the density function that is just 1.1158
DY1, that is the inside one, and DY2.1165
Notice here that, the function F = Y1 that is what I'm integrating.1170
That is positive on the right hand triangle and negative on the left hand triangle.1176
It is symmetric, and the shape of the region we are integrating over is symmetric too.1181
This double integral is equal to 0, by symmetry.1188
This thing is completely balanced around the Y2 axis.1193
I can work out that double integral, it is not a very hard double integral.1200
But, I'm feeling a little lazy today and I do not think I want to work that out.1204
I'm just going to say by symmetry it is equal to 0, if I did work out that integral.1209
It might not be a bad idea, if you are little unsure of this to work out the double integral and1214
really make sure that you get 0, so that you believe this, if it seems a little suspicious to you.1219
In the meantime, I will go ahead and calculate the expected value of Y2.1226
Same double integral, at least the same limits, Y2 = 0, Y2 = 1, and Y1 = Y2 -1, and Y1 = 1 - Y2.1230
I'm finding the expected value of Y2, I do Y2 × 1 DY1 DY2.1248
Y2 is not symmetric over this region, because it ranges from 0 to 1.1258
I can not get away with invoking symmetry, again.1264
I have to actually do this double integral.1268
I will go ahead and do it, it is not bad.1273
I'm integrating Y2 with respect to Y1.1275
Y2 is just a constant, that can give me Y2 × Y1.1278
I'm integrating that or evaluate that from Y1 = Y2 - -1 to Y1 = 1 - Y2.1285
I get Y2 × 1 - Y2 - Y2 -1 - (Y2-1).1296
That is Y2 ×, it looks like -2Y2 -2.1314
That is -2Y2² -2Y2².1322
That was all just solving the inside integral, I need to integrate that from Y2 = 0 to Y2 = 1.1331
This is all integrating DY2.1349
I think I screwed up a negative sign in there.1357
It is not -2, it is +2.1361
That +2, when I multiply it by Y2, I made a couple of mistakes in there.1365
That is 2Y2 - 2Y2², I think that is correct now.1372
Let me go ahead and integrate that from Y2 = 0 to Y2 = 1.1379
That looks like I'm spacing out on that line right there.1386
I think I got it right now.1390
I want to integrate that, the integral of 2Y2 is Y2².1392
The integral of 2Y2² is -2/3 Y2³.1400
I need to evaluate this from Y2 = 0 to Y2 = 1.1410
If I plug in Y2 = 1, I will get 1 -2/3, plug in Y2 = 0, I just get nothing.1417
I get 1 -2/3 is 1/3, that is my expected value for Y2.1428
I should have boxed in my answer for expected value of Y1, because that was my first answer up there.1434
I had a couple of hitches along the way there, but I think it worked out okay.1441
I still need to find the expected value of Y1 Y2.1446
Again, that is that same double integral Y2 = 0 to Y2 = 1 and Y1 = Y2 -1 to Y1 = 1 - Y2.1452
Where is my function, it is Y1 × Y2 × the density function which is just 1 of DY1 DY2.1470
If that is getting a little cut off there, then I will just say that that last symbol was DY2.1483
In case, you have trouble reading that.1489
This is not as bad as it seems, because the function that I'm integrating Y1 Y2,1493
it is going to be positive on the right hand triangle and negative on the left hand triangle.1500
They are exactly evenly balanced, is symmetric on this triangle.1506
That means that whole integral will come out to 0, by symmetry.1521
That is because, I had a function that is positive on the right hand part and negative on the left part.1528
If that feels a little suspicious to you, go ahead and do the integral.1534
It will be a little messy, it is likely the most fun integral in the world but it is possible.1538
It is just tedious algebra, there is really nothing too dangerous in there.1542
You can do the integral and you should get 0, at the end.1547
It should agree with what I got by symmetry.1550
That completes that problem, we will be using the same setup and the same values for example 2.1555
I want to make sure that you understand everything we did here,1565
because I'm going to take these answers and I'm going to use them for example 2.1568
Just to make sure that you are very comfortable with all this stuff,1574
before we move on and use these answers in example 2, let me recap the steps here.1577
I wanted to look at this triangle.1583
First of all, I graphed this triangle based on those 3 corner points that were given to me.1586
I set up this triangle and colored in the region here.1592
I know that I'm going to be doing a double integral, I was trying to describe the limits of this triangle.1597
If I list Y2 first then I can do it just by doing one double integral.1606
If I list Y1 first, then I have to chop this thing into two.1612
That is why I listed Y2 first going from 0 to 1.1616
And then, I had to describe Y1 in terms of these two lines.1619
The way I got those two lines was, I found the equations of these two diagonal lines on the sides of the triangles.1624
Then, I solve each one for Y1.1632
That is where those limits right there came from, in terms of Y1.1635
That let me set up my limits of integration on each of my three integrals.1639
Those sets of limits, I use the same sets of limits on each one of those integrals.1644
Each one of those integrals, I integrated something different.1655
They all had this 1 in them, and that 1 came from this one right here, the density functions.1658
That 1 manifested itself there, there, and there.1664
I had the 1, then each one I was integrating something different because1669
I was trying to find the expected value of something different.1672
Expected value of Y1, integrate Y1.1676
Expected value of Y2, integrate Y2.1678
Expected value of Y1 Y2, integrate Y1 Y2.1682
I know I have set up three integrals and two of them, I notice that the function I'm integrating is symmetric,1687
in the sense that Y1 is going to be positive over here and negative over on this region.1694
It exactly bounces each other out, I know that I'm going to get 0, if I do that integral.1701
If you do not believe that, just do the integral, it would not be that hard.1707
It should work out to be 0.1710
Same thing over here, Y1 Y2 is positive in the positive region, negative in the second quadrant.1711
It is going to give me 0, by symmetry.1720
That is where I got these two 0's from.1722
If you do not like it, just do the integrals and you can work them out yourself.1725
The expected value of Y2, Y2 is positive on both of those triangles.1729
I cannot use symmetry, I actually have to do the integral.1734
I integrated with respect to Y1, I got Y2 Y1.1738
Plugged in my limits, I got an integral in terms of Y2.1741
Integrated that, plug in my limits and got the expected value of Y2 to be 1/3.1746
Hang onto these 3 answers, we are using it again right away in example 2.1752
In example 2, we have the same setup from example 1.1758
Let me go ahead and draw out that setup.1762
It was this triangle with corners at -1,0 and 0,1 and 1,0.1764
Let me draw that triangle, it is the same thing we had in example 1 and the same density function.1773
A joint density function is always equal to 1 over that region.1782
What we want to do here is, we want to calculate the covariance of Y1 of Y2.1789
After that, we are going to ask whether Y1 and Y2 are independent.1795
I'm going to use that formula that I gave you for covariance of Y1 and Y2.1799
Not the original definition which I think is not very useful, is often cumbersome if you use the original definition of covariance.1804
We have this great formula for covariance.1813
What covariance of Y1 and Y2, is equal to the expected value of Y1 × Y2 - the expected value of Y1 × the expected value of Y2.1815
You cannot necessarily separate the variables, it is not necessarily true that1831
the expected value of Y1 × Y2 is equal to the expected value of Y1 × the expected value of Y2.1836
The reason this formula is useful for us right now is,1845
we already figured out each one of these quantities in example 1.1848
The work here, it was quite a bit of work was done here in example 1.1856
If you did not just watch example 1, if example 1 is not totally fresh in your mind, just go back and watch it right now.1862
You will see where we work out all of these quantities individually.1869
The expected value of Y1 × Y2 was 0.1874
The expected value of Y1 was also 0.1878
The expected value of Y2 was 1/3.1881
That is what we did in example 1.1884
This all simplifies down to be 0 here.1886
That is our answer for the covariance, the covariance is 0.1890
If Y1 and Y2 are independent, this is a very subtle issue here because you see that1896
the covariance being 0 and you might think of independence, that is the converse of our theorem.1902
Our theorem says that, if they are independent then the covariance is 0.1909
It does not work the other way around.1915
We cannot say necessarily that they are independent yet.1918
In fact, I want to remind you of a theorem that we had back in another lecture on independent variables.1921
From the lecture on independent variables, that was several lectures ago.1934
You can go back and check this out, if you do not remember it.1942
We have a theorem that the region must be a rectangle.1945
And then, there was another condition that the joint density function must factor into two functions.1964
One just of Y1 and one just of Y2.1975
If both of those conditions were satisfied then the variables are independent, that was if and only if.1981
In this case, we do not have a rectangle.1989
This is not a rectangle, it is a triangle.1993
That theorem tells us that Y1 and Y2 are not independent.2004
That should kind of agree with your intuition, if you look at the region, if I tell you for example that Y1 is 0.2018
Let me put some variables on here.2029
This is Y1 on the horizontal axis and this is Y2 on the vertical axis.2031
If I tell you that Y1 is 0 that means we are on this vertical axis, then Y2 could be anything from 0 to 1.2036
If I tell you that Y1 is ½ then Y2 cannot be anything from 0 to 1, it could be only as big as ½.2053
By changing the values of Y1, I'm changing the possible range of Y2,2064
which suggests that knowing something about Y1 would give me new information about Y2.2071
That is the intuition for dependents, for variables not being independent.2077
You should suspect, by looking at that region, that the variables are not independent.2086
This theorem from that old lecture confirms it.2092
Just to recap here, we are asked to find the covariance.2097
I'm using the formula that I gave you for covariance.2101
I think it is on the first or second slide of this lecture, just scroll back and you will see it,2105
the definition and formulas for covariance.2111
It expands out into these three expected values.2113
We calculated all these in example 1.2117
I just grabbed the old values from example 1 and simplify down, just got 0,2119
which might make you think independent because we had this theorem that if they are independent, then the covariance is 0.2125
The converse of that theorem is not true.2133
This is kind of the classical example of that.2137
Variables can have a covariance equal to 0 and not be independent.2141
This is really what this example is showing, not be independent.2157
It is true that if their independent then the covariance is 0, but they can have covariance 0 and in this case,2161
since the region is not a rectangle, they are not independent.2168
In example 3, we have independent random variables.2176
We have been given their means and their variances.2179
We are given a couple of linear functions U1 is Y1 + Y2 and U2 is Y1 - Y2.2182
We want to calculate the variance of U1 and U2.2190
Let me show you how those work out.2195
The variance of U1, U1 is, by definition is Y1 + 2Y2.2197
I'm going to use the theorem that we had on linear functions of random variables.2212
This was on the introductory slides, you can go back and just read back about that theorem,2221
about linear functions of random variables.2229
What that told me is that, a linear combination of random variables distributes out2232
but you write the coefficients 1² × σ 1², the variance of Y1.2238
Let me write that as the variance of Y1, 1² × the variance of Y1 + 2² × the variance of Y2.2247
And then, there is this other term which is kind of obnoxious but we have to deal with it.2257
It is 2 × the coefficient 1 × 2 × the variance of Y1 with the covariance of Y1 with Y2.2261
That is by the theorem from the beginning of this.2281
I think the title is linear functions of random variables.2290
That was the theorem that we had.2296
Let me plug in, 1² is just 1, the variance of Y1 we are given is 4 + 2² is 4.2299
The variance of Y2 was given to be 9 + 4.2306
The covariance of Y1 × Y2, what we are given is that Y1 and Y2 are independent.2313
If they are independent, then the covariance is 0.2320
Remember, in example 2, we learned the converse is not true.2324
But, it is true that if they are independent then the covariance is 0.2326
That right there is by independence and by the theorem that we learned earlier on in this lecture.2331
We just simplify, 4 + 4 × 9 is 4 + 36 which is 40.2341
That is the variance of U1, the variance of U2 is, by definition U2 is Y1 - Y2.2349
I will just do the same thing, I'm going to expand it out using that theorem.2360
We got 1Y1, 1² × the variance of Y1 + -1² × the variance of Y2, I'm using that theorem,2365
you have to square the coefficients.2375
+ 2 × 1 × -1, those are the coefficients × the covariance of Y1 with Y2 and 1.2378
We just get 1 × the variance of Y1 which were given is 4 +, because -1² gives us a positive then Y2 is 9.2390
Again, the covariance of Y1 Y2, by independence is 0.2403
This just simplifies down to 13 is the variance of U2, and we are done.2408
To recap the steps there, the variance of U1, expand that out.2416
U1 was defined to be Y1 + 2Y2.2420
We had this theorem on linear functions of random variables.2425
Go back and check it out, it told us how to find the variance of a linear combination.2428
You kind of expand out the variances, you have to square the coefficients.2435
That 1² comes from the sort of a hidden 1 right there, and that 2² comes from there.2439
This 2 comes from the theorem.2448
This 1 and this 2 come from the coefficients, there and there.2451
This covariance comes from the theorem as well.2456
We are given that Y1 and Y2 are independent, independence tells us that their covariance is equal to 0.2464
That was a theorem that we also had earlier on in this lecture.2471
The variance of Y1 is 4, that came from here.2476
The variance of Y2 is 9, that came from the stem of the problem here.2481
We drop those in, simplify down, and we get 4 + 4 × 9 is 40.2486
The variance of U2 works exactly the same way, except that there are coefficients now.2493
Instead of being 1 and 2, are 1 and -1.2498
The covariance drops out because they are independent.2501
We get 4 and 9, remember the -1 squares because you always square the coefficients with variance.2505
That makes a positive and 4 + 9 is 13.2512
By the way, we are going to use these values again, in example 4.2517
Make sure you understand these values very well, before you move on to example 4.2525
We will need them again.2531
Example 4, we are going to carry on with the same sets of data that we had from examples 3.2534
Y1 Y2 are independent variables, we have been given their means and their variances.2540
We are going to let U1 and U2 be these linear combinations.2547
I want to find the covariance of U1 and U2.2551
And then, I'm also going to find this ρ here, is the correlation coefficient.2557
We learn about that, at the beginning of this lecture.2561
Let me expand out the covariance of U1 and U2.2565
U1 and U2 is the covariance of Y1 + 2Y2 and Y1 - Y2, that is just the definition of U1 and U2.2569
And then, there was a theorem that we had to back at the beginning of this lecture2584
on how you can expand covariances of combinations of a random variables.2590
It says, it expands out linearly so it kind of expands out.2598
You can almost think of foil, first outer, inner last.2604
It is the covariance of Y1 with Y1 - the covariance of Y1 with Y2 + 2 × the covariance of Y2 with Y1.2608
I'm just expanding it out, first outer and inner last, -2 × the covariance of Y2 with Y2.2629
It is just expanding out each term with each term.2640
We are going to use a couple other facts here.2645
Remember that, the covariance on Y1 with itself is just the variance of Y1.2647
That was something we learned on the very first slide of this lecture,2653
it was the one where we introduced covariance.2660
I gave you, first the definition of covariance and a couple useful formulas.2663
One useful formula was the covariance of a variable itself is just the variance.2667
We also have theorem that said, if Y1 Y2 are independent and they are given to be independent here,2672
then the covariance is 0.2679
This is 0 by independence.2681
This covariance is also 0, by independence.2689
Covariance of Y2 with itself is the variance of Y2, that same formula that we had on the first slide.2693
This is the variance of Y1 -2 × the variance of Y2.2703
Where is the variance of Y1, there it is, it is 4 -2 × the variance of Y2 is 9.2708
That is 4 -18 which is -14, that is our covariance.2717
We also have to find the correlation coefficient.2726
Let me remind you that the correlation coefficient, how that is defined.2729
The correlation coefficient, ρ of U1 U2, by definition is the covariance of U1 with U22734
divided by the standard deviation of U1 and the standard deviation of U2.2750
I'm going to use σ for the standard deviation of U1 × the standard deviation of U2.2758
That σ is the σ for U1, it is not the σ for Y1.2764
It is not the σ 1 right here, that σ U2 is not the σ 2 there.2769
That was a mistake that I accidentally made, when I was making a rough draft of these notes.2776
Do not make the same mistake I did.2781
The covariance of U1 U2 is –14, we just figure that out.2785
The standard deviation of U1, I'm going to invoke what I figured out on the previous example.2790
The standard deviation of U1 is just the square root of the variance of U1, and the same for U2,2801
the square root of the variance.2810
The standard deviation is always the square root of the variance, that is the definition.2811
I figured out the variance of U1 and U2, in the previous example, in examples 3.2817
I'm trying to look up those values right there.2826
We work this out in examples 3.2831
We solved -14 in the numerator.2839
In example 3, we figure out that the variance of U1 was 40, we have √ 40.2841
The variance of U2 was 13, that is what we figured out in example 3.2848
If you did not just watched example 3 and you think those numbers appeared magically,2856
go back and watch example 3, and you will see where they came from.2860
This does not get much better, this is -14, √ 40 we can pullout a 4, make that 2 √ 10.2864
√ 13 in the denominator and that turns into, if we cancel 2, we get -7/10 × 13 is 130.2874
Not a particularly nice number, I could not find figure out a way rake up these numbers to behave nicely.2886
I did throw this into a calculator and what did I get there, when I plugged in a calculator.2892
This was 0 × 0.614, actually that was an approximation and it is negative.2902
That is what I got when I plug that number into a calculator, nothing very revealing there.2911
Let me mention that one thing we knew about the correlation coefficient,2917
I gave you this way back on the third slide of this lecture.2923
When I talked about correlation coefficient is that, the whole point of correlation coefficient is its scale independent.2929
It is always between -1 and 1, and this is between -1 and 1 because it is -0.6.2940
That is slightly reassuring, if it had been outside of that range then2949
I would have known that I have made a mistake, somewhere through here.2955
The fact that I got a number in between -1 and 1, does not guarantee that right2958
but it makes me feel a little more confident in my work here.2962
We got answers for both of those, let me show you how I got those.2966
The covariance of U1 and U2, I expanded out the definition of U1 and U2 which is Y1 + 2Y2 and Y1 -Y2.2970
We had this theorem on one of the introductory slides to this lecture which said,2982
how you can expand out covariances of linear combinations.2987
It is very well behaved, it just expands out the same way you would multiply together binomial.2991
You can think of foiling things together.2996
We did the first Y1 and Y1, the outer Y1 and Y2, subtracted because of the coefficient there.2999
Let me write out foil here, if you remember your high school algebra, first outer, inner last.3008
The inner term is 2Y2 × Y1, the covariance of it and the last term is -2Y2 Y2.3013
These mixed terms, the Y1 and Y2, remember they were given that the variables are independent.3023
Independent variables have covariance 0.3030
That is not true in the other direction, just because they have covariance 0, does not mean they are independent.3033
If they are independent, then the covariance is definitely 0, which is great, those two terms dropout.3039
The other thing we learned by an early formula, when I first taught you about covariance3045
is that the covariance of Y1 with itself is just the variance of Y1 and same thing for Y2.3052
I can just drop in my value for the variance, there it is right there.3061
The variance of Y1 is 4, variance of Y2 is 9, and drop those in and simplifies down to -14.3064
The correlation coefficient ρ, by definition, is the covariance of those two variables divided by their standard deviations.3074
I just figure out the covariance, that is the -14.3083
The standard deviations are always the square root of their variances, that is the definition of standard deviation.3086
Take their variance and take the square root.3093
The variance of U1 and U2, that is what I calculated back in example 3.3096
Just go back and watch example 3, you will see where these numbers 40 and 13 are coming from.3103
It is not these numbers right here, the σ 1² and σ 2² because those were the variances for Y1 and Y2.3109
Here, we want the variances of U1 and U2.3118
Once, I got those numbers in there, I reduce the square roots a little bit, but it did not end up being a very nice number.3122
The reassuring thing was that when I found the decimal, it was between -1 and 1,3129
which is the range that a correlation coefficient should always landed.3135
Last example here, we got independent variables but they all have the same mean and the same variance.3141
We want to find the mean and the variance of the average of those variables.3148
The average just means, you add them up and divide by the number of variables you have.3152
It is 1/N Y1 up to 1/N YN.3158
I wrote it that way to really suggest that, that is a linear combination of the original random variables.3162
This is a linear function and we can use our theorem on how you calculate means and variances of linear combinations.3169
That is what I'm going to use.3183
The expected value is the same as the mean.3185
Remember, the expected value of Y bar, the mean and variance of the average.3188
The expected value of Y bar is just the expected value of 1/N Y1 up to 1/N YN.3195
I can distribute by linearity of expectation, that was a theorem that we had.3208
I can distribute and pull out those coefficients 1/N × E of Y1 up to 1/N × E of YN.3217
That is 1/N × μ up to 1/N × μ.3231
They all have the same mean μ.3236
If you add up N copies of 1/N × μ, you just get a single copy of μ.3239
That is my expected value of the average.3249
The variance of the average, variance of Y bar, again is the variance of 1/N Y1 + up to 1/N YN.3252
Variance is not linear, expectation is linear.3265
There is a nastier theorem that tells you what to do with variance.3270
I gave you that theorem in one of the introductory slides.3277
I said linear functions of random variables.3280
The way this works is, you pull out the coefficients but you square them.3284
1/N² × the variance of Y1 up to 1/N² × the variance of YN.3290
There is the cross terms, there is this cross term which is 2 × the sum as i is bigger than j3300
of the coefficients 1/N × 1/N × the covariance of YI with YJ.3310
That looks pretty dangerous there but, let us remember that we are given that we have independent variables.3321
Any Y and J, if you take their covariance, since they are independent, this covariance will be 0, by independence.3331
That is a really nice, that means I can just focus on the first few terms here, 1/N²3342
the variance of Y1 is σ² + up to 1/N² × σ².3350
Let me write that a little more clearly, that is 1/N² in the denominator there.3360
What I have is N terms here of 1N² × σ².3368
That simplifies down to σ²/N.3374
By the way, this is a very fundamental result in statistics.3379
This is something that you use very often, as you get into statistics.3385
The variance of the mean is equal to σ² divided by N.3390
This is where it comes from, this is where the magic starts to happen is right here with this example.3395
Let me make sure that you understand every step here.3402
We want to find the expected value of Y bar.3405
Remember that, Y bar the average is just can be written as a linear combination of these variables.3408
Expectation is linear, that is what we learned in that theorem.3415
You can just separate it out into the expected values of Y1 up to YN, then just pull out the constants.3419
And then, each one of those E of Yi is μ because we are given that in the problem right there.3426
We are adding up N copies of 1/N × μ.3437
At the end, we just get a whole μ.3441
The variance is a little bit messier, the variance of a linear combination.3443
Again, you can split up into all the separate variances but when you pull it out, pull out the coefficients, it get².3450
That is why we get 1/N² on each of these coefficients.3457
There is this cross term, 2 × the sum of the coefficients.3462
That is a little messy there but that is 1/N × 1/N.3467
That is coming from these coefficients right here, the covariance of Yi × Yj.3471
The fortunate thing is that, we have a theorem that says when two variables are independent, their covariance is 0.3480
Converse of that is not true, it could be covariance is 0 without independence.3488
But if they are independent, their covariance is definitely 0.3493
We are given that they are independent here.3497
All those cross terms dropout and we are just left with N copies of 1/N².3500
We are given that the variance of each individual variable σ².3507
N × 1/N² is just 1/N, we still have that σ².3512
The variance of the mean is σ²/N.3518
Essentially, this means if you take the average of many things,3522
it is not going to vary as much as individual members of the population will,3525
because the variance shrinks down, as you take a larger and larger sample.3531
That is a very classic result in statistics, now you know where it comes from.3537
Now, you know where this classic formula comes from.3542
That wraps up our lecture, a kind of a big one today on correlation and covariance, and linear functions of random variables.3546
This is all part of the chapter on Bivariate distribution functions and Bivariate density functions.3554
In fact, this wraps up our chapter on Bivariate density functions and distribution functions.3562
We will come back later and talk about distributions of random variables.3568
We still have one more chapter to go.3573
In the meantime, it is nice to finish our chapter on Bivariate density and distribution functions.3575
This is all part of the probability lecture series here on www.educator.com.3581
I'm your host and guide, my name is Will Murray, thank you for joining me today, bye now.3586


text:
I'm excited to have Bill Coleman on the show Thursday. Bill was the photographer for Amish Peace. He is AMAZING with a camera. Be sure to visit his website and look through his beautiful photographs. Breathtaking. Truly. Join us tomorrow at 4pm Central here: http://toginet.com/shows/amishwisdom (to listen just click the player in the top right corner.)
Bill Coleman is an internationally acclaimed photographer of the Amish. The photographic legacy he deems his Amish Odyssey is four decades long, a goodly number of years to document the lives and landscapes of the Old Order Amish sect in a hidden valley somewhere in Pennsylvania. But forty years is scantly time to portray a people whose ancestors immigrated to the area in the 18th century.
The Amish have a love affair with life and simplicity, with preserving ages of innocence and respect, with teaching responsibility and the joys of tradition. These very private people are reluctant to reveal themselves to outsiders.
Bill, albeit knowing he will always be an outsider has, charmingly, respectfully cajoled his way into the hearts and homes of a number of Amish families willing to allow him to document their lives from birth to death, generation after generation. His determination to keep their location secret, their lives private, equals their determination to maintain their way of life.
His photographs, often images of a precise scene taken at each season of the year and of cherubic children as they mature, holds a continuity of life and lifestyle beholden to the past, yet treasured for the future. In a sect where tradition and the ethic of hard work, respect of elders and love of children intertwine.
Bill creates beautifully powerful images. The past is present in every nuanced curve of the land. His photographs are intimate portraits of a place and a people who live in the simplicity of the past as the realities of the 21st century encroach upon their future.
Bill's photos touch one's senses beyond the gift of sight. They reveal the laughter of the youngsters, the chill of a snowy field, the aroma of a fresh baked apple pie, the touch of little girls clutching their baby dolls, and the mellifluous hammering of a barn raising. Beyond the physical senses are those of place, of humor, of past, and present thriving harmoniously. He portrays the whimsy of children at play with equal respect to the seriousness of Sunday worship.
On many days, from sunrise to sunset, Bill aims his camera at what he calls the "symphony of forms that created the valley…the lights, colors and textures of the fields, the farmlands, and the faces." In describing his passion for preserving the gentle nature of this Amish sect, he confides, "I wish they knew how much I love them." Look at his photos and you will have no questions about his love.
For more information and see some his amazing work, please visit: www.amishphoto.com
WIN A SIGNED COPY OF BILL'S LATEST BOOK, THE GIFT OF FRIENDSHIP! LEAVE A COMMENT on Bill's Amish Wisdom segment post (along with your email!)

subset: commoncrawl


text:
A well-rested Hyun-jin Ryu gave the Dodgers another strong performance on Tuesday night, and a pair of mammoth solo home runs were enough to beat the Giants 2-1 in the opening contest of a three-game series at AT&T Park in San Francisco.
Ryu, who last pitched on Sept. 16 in Phoenix, allowed just one run in seven innings of work for his 14th win of the season. On seven days rest Ryu had six strikeouts and lowered his ERA on the season to 2.97. Since the All-Star break he has just nine total walks in 11 starts, including one on Tuesday.
Ryu was backed by two solo home runs. Yasiel Puig hit one 436 feet to the deepest part of right center field in the fifth inning, and Matt Kemp hit one 443 feet to left center field in the sixth.
For Puig, it was his 19th home run of the season, the fifth most by a Dodgers rookie since the team moved to Los Angeles in 1958. Puig has hit home runs in three of his last five starts and has six home runs in September.
Kemp was 2-for-4 on the night and since returning from the disabled list on Sept. 16 is 8-for-24 with a pair of doubles, hitting .333/.385/.542.
The only run for the Giants also came via home run, and even that was hit by someone originally signed by the Dodgers. Old friend Tony Abreu went deep in the fifth inning, one of just four hits allowed by Ryu on the night.
Former Giants closer Brian Wilson pitched his first game in San Francisco since April 2012, and after he was greeted with mostly boos he pitched a perfect eighth inning, including a pair of called strikeouts. Wilson as a Dodger has allowed one run in 16 appearances, with 12 strikeouts and three walks in 12⅓ innings.
Buster Posey singled against Kenley Jansen in the ninth inning and as the tying run advanced to second base on a wild pitch (that probably should have been called a passed ball) with just one out. But Jansen struck out Hunter Pence and Pablo Sandoval to end the game for his 28th save.
- With the win the Dodgers are now 8-9 against the Giants this season, including 3-4 in San Francisco.
- Jansen now has 109 strikeouts on the season, one shy of Aroldis Chapman for the major league lead among relief pitchers.
- Carl Crawford entered the game in a 5-for-42 slump of his own but had three hits, including a double, and a stolen base on t.he night. It was Crawford's 12th game of the season with at least three hits.
- Jerry Hairston struck out as a pinch hitter in the ninth inning and is suffering through slumps of 0-for-16 and 3-for-42.
Up next
The Dodgers will try to get Ricky Nolasco back on track on Wednesday night in the middle game of the series. The Giants originally planned to go with Madison Bumgarner on Wednesday but have instead decided to shut down the left-hander for the season after 201⅓ innings pitched in 2013. Barry Zito, who last pitched on Sept. 2, will start instead for San Francisco.
Tuesday particulars
Home runs: Yasiel Puig (19), Matt Kemp (6); Tony Abreu (2)
WP - Hyun-jin Ryu (14-7): 7 IP, 4 hits, 1 run, 1 walk, 6 strikeouts
LP - Matt Cain (8-10): 7 IP, 7 hits, 2 runs, 3 walks, 4 strikeouts
Sv - Kenley Jansen (28): 1 IP, 1 hit, 2 strikeouts


text: Ring made out of 9K gold and enamel in aqua blue. Statement ring with length, best worn on the middle or index finger. Linearity in shape, color tensions and the reflective properties of enamel are its defining elements. A jewel essential for every bold stylistic look that will not go unnoticed.


text:
How do we keep learning?
When I first entered the risk management profession, I had the benefit of working with a number of wiser and more experienced practitioners. I think that experience is far more common than not in our line of work, and nothing brings that home to me more than attending a conference. At these gatherings, risk and insurance professionals of all levels of experience come together for one common goal, to share knowledge. The speakers are obviously there to share, but more than that, the attendees are there to share as well. Much of that occurs at the less formal settings , at the lunches and receptions, at the dinners and in the hallways. It's in those less structured venues that some of the most important work of a conference happens. Of course, what I'm referring to is relationships; building, reinforcing, renewing them. In many ways, that is the most important work that conferences can contribute to our profession. Relax into the flow of the conference, and keep those relationships growing.
From the 2018 IRMI Construction Risk Conference


text:
The National Pilgrim Virgin Statue and Relics will arrive at Westminster Cathedral on February 18
Cardinal Nichols will crown Our Lady of Fatima and consecrate England to the Immaculate Heart of Mary during a Mass at Westminster Cathedral on Saturday February 18.
The consecration will be a renewal of the original offering made by Cardinal Bernard Griffin on July16 1948.
The National Pilgrim Virgin Statue and Relics will arrive at 1.45pm on Saturday and remain in the Cathedral until 2.30pm the following day.
After Mass people will be invited to venerate the relics and participate in Marian devotions. This will be followed by a Fatima Symposium in Westminster Cathedral Hall.
2017 marks the 100th anniversary since Our Lady first appeared to two children in Fatima, Portugal.


text:
Some inside baseball that's relevant to the bum's rush (Uncle's rush, really) toward an Electric Car Future:
They don't send me electric cars to test drive. Not one. Not yet. Probably not ever.
They send me everything else – except for GM, which stopped sending me cars to punish me for expressing un-PC opinions about the company's leg-humping of "diversity" (see here for more about that). But the point is, GM did send me cars and could send me cars.
Why is this relevant to the EV discussion?
Because it demonstrates badly how gimped EVs are – and I figured someone (it won't be the "consumer" press) ought to tell you about it, in case you've taken too many pulls on the electric car crack pipe and actually bought into the EV hype, which is potentially much more dangerous to your wallet than a time share or a Pampered Chef distributorship.
Okay, gather 'round.
My place in SW Virginia is about 200 miles from the central press car hub for the northeast region, which is located in the Northern Virginia/Maryland suburbs. Press cars are brand-new cars put into a special fleet for distribution to car journalists like me. A driver who works for the fleet management company drives a new car to a journalist's house or place of business, drops it off for him to test out for a week and then comes back the following week to pick it up.
It's no problem for the delivery drivers who work for the press fleets to make the trip from there to here in 3-4 hours, traffic depending. Usually doable on a full tank but if it's a muscle car or some such, no problem. Just refuel and back on the road.
No worries.
Unless it's an EV.
Because an electric car can't make it from there to here without stopping at least once – for a minimum of 30-45 minutes each time and that's the best-case scenario, assuming the driver can find a "fast" (high voltage) charger. These are still hard to find – and if one can't be found, then it's an overnight stay for the driver and what was a 3-4 drive becomes a two-day ordeal.
Two days' pay for the driver, plus the cost of a hotel for the overnight stay. The EV will not have burned any gas during the odyssey but a lot money and time will have been burned through.
The other option is a flatbed truck – which could get the EV here in a few hours – but that gets into money (and time) too. And as hard as it may be for government parasites to grok, people who can't legally just steal money to support their activities have to think about what things cost and operate within certain economic parameters.
So, no EVs for me – or any other car journalist outside the orbit of an EV's range on a single charge. Which means only urban/suburban journalists within about 50 miles of the press car hub get EVs to test drive.
This accounts for the vacuous, dishonest coverage of EVs. All you need to do is flip it around to see what I mean.
Imagine Consumer Reports or some other Pravda-like organ of the establishment car press got a non-electric car to evaluate and it conked out after 100 miles (or even 200 miles) and had to be hooked to a special and hard-to-find apparatus for at least 30-45 minutes before they could it back on the road again.
Imagine this car's performance noticeably deteriorated in cold weather; that it could not be refueled in very cold weather – unless an enclosed/heated garage could be found. That running the car's heater and headlights had the same effect on the car's range as a hole in the gas tank of a non-EV would have.
The resulting coverage would make what happened to Ford/Firestone back in the '90s seem like a minor bitch about the location of the cupholders. There would be open talk of gyps and cons and general preposterousness. A car that can't make a 200 mile trip without a major (and possibly overnight) pit stop?
That is something even a Yugo could do.
And the Yugo didn't cost what an EV costs.
But you'll not be told such tales by the Pravda-like car press because the only car press that gets EVs is the urban/suburban press and that is very much like expecting to get political coverage that's even-handed toward Trump out of a Latinix transqueer gender studies professor at Berkeley.
Which brings to me to the worshipful coverage given the 2019 Audi eTron – which is Audi's first all-electric crossover SUV and the tip of the spear – so to speak – of what is on deck to become an LBGTSUV EV juggernaut.
It is a very weird juggernaut in that there is almost no market demand for these EVs, which in total comprise barely 1 percent of all new cars sold, almost all of them in urban areas of California and Arizona (where, not coincidentally, it is warm all year long). Yet Audi (and VW) as well as Mercedes and BMW are practically tripping over each other to get as many EVs into production as possible.
Who do they think is going to be buying them?
The eTron will sticker for $75,000 – to start. A "well-equipped" model will sticker for $86,700.
Keep in mind that Audi will be competing (if that's the right word) for buyers in that 1 percent bracket with all the other hawkers of similarly high-dollar EVs. If the market for these EVs doesn't somehow come into existence after the fact, there is is going to be much more than Hell to pay. It could mean the collapse of several big-name car brands, who've "invested" (abuse of language, hence air quotes) in vehicles for which there is no market, in the desperate hope that one will come into being.
It is not a coincidence that the biggest bum's rushers toward the Electric Car Future are German companies – Audi/VW, BMW and Benz. It is because in the home market there are fatwas even more severe and lunatic than those in force here. Fatwas which forbid cars that aren't capable of at least part-time all-electric operation from even being used in certain "protected" areas, such as urban areas. These areas are being expanded upon. It is an artificial – government created – "incentive" to build EVs, though no one seems to ask aloud what "incentive" buyers will have to purchase them if they can't afford them.
It doesn't matter. The car companies won't contradict the now-sacrosanct EV orthodoxy, even if it ultimately kills them. The reason being that management isn't going to suffer. The fish rots from the head down, but in the case of huge corporations, the heads are extremely well-paid even if the company rots under their watch, so what do they care? Most of them are not "car guys."
They are money guys (and gals).
And so long as the getting's good, they'll be getting – and then, going.
Meanwhile, the Pravda press will continue on its knees, supplicating the EV god.
And outliers like me will sit back and watch the show.
. . .
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Our PHP 1 Click Framework Installer was created to enable you to use full PHP frameworks for your web site, with absolutely no setup required from you. You no longer need to seek out the most current release of a framework for your enterprise and dedicate many hours on downloading as well as setting it up. Now you have everything available inside your Website Control Panel, displayed in a simple to use interface.
Different frameworks accessible
Commence your web enterprise using a framework
From the HOSTINGFROM.COM Website Control Panel, you'll be able to to add a favored open–source PHP framework. We've prepared a range of the most popular frameworks that are now available on the web – CakePHP, Laravel, CodeIgniter, Zend, Symfony, and YII. You'll be able to install every framework on any website that is hosted inside your hosting account.
Our PHP 1 Click Framework Installer tool can be acquired with all of the web hosting solutions which feature the Website Control Panel – cloud hosting plans, VPS web hosting plans,dedicated web hosting plans, and semi-dedicated hosting plans.
1–click PHP framework setting up
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The PHP 1 Click Framework Installer is meant to work on the very same basic principle like our App Installer – through minimum effort from you and the system doing all of the hard work on your behalf. PHP frameworks are installed with just a number of mouse clicks with no configuration required from you. All you need to do is select the place of your PHP framework. That's all.
We keep a log of the installed PHP frameworks and you can un–install every PHP framework which you don't need with simply a click of the mouse.
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The easiest approach to back up a framework
When your information is online, the easiest way to guard it is without a doubt by keeping a backup. You can expect routine data backup for every one of the cloud hosting plans accounts. Nonetheless, if you'd like to be in control, you can back up your frameworks at any moment. We've produced an intelligent system that lets you generate a backup with a click.
There is no limit for the amount of backup files you can create or on how often you may create them. All of the backups will be placed in a directory within your account. When you have free space, you may create as many completely new backups as you need.


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Ultimate Community Knowledge Quiz
Community got its second life once it premiered on Netflix. Worldwide audiences have a chance now to get to know the amazing study group and their adventures. And who knows? Maybe Netflix will chip in and give us the thing that all fans are waiting for – "the movie". That would be a dream come true for every fan of the show. But for now, all we have are constant rewatches. To celebrate this amazing show we preapred this 20 questions hard quiz about Community. Will you be able to ace it? Test your knowledge and see if you really know Community as well as you think you do. If you like this quiz, please share it, because it helps us a lot and allows us to continue our work on this site. We hope to show you some more upgrades in the future so if you are a regular, stay tuned! Comment on how well you did so others can compare their scores to yours to see who knows Community best. Good luck and have fun!
ABOUT Community:
Community is an American comedy television series created by Dan Harmon that aired on NBC and Yahoo! Screen from September 17, 2009, to June 2, 2015. The series follows an ensemble cast of characters played by Joel McHale, Gillian Jacobs, Danny Pudi, Yvette Nicole Brown, Alison Brie, Donald Glover, Ken Jeong, Chevy Chase, and Jim Rash at a community college in the fictional town of Greendale, Colorado. It makes heavy use of meta-humor and pop culture references, often parodying film and television clichés and tropes.
Harmon based the program on his own experiences attending a community college. Each episode was written in accordance with Harmon's "story circle" template, a method designed to create effective, structured storytelling. Harmon served as the series' showrunner for its first three seasons, but was fired prior to the fourth and replaced by writers David Guarascio and Moses Port. After a lukewarm response to the fourth season from fans and critics, Harmon was re-hired for the show's fifth broadcast season, after which it was canceled by NBC. Yahoo! Screen commissioned its sixth and final season, which concluded on June 2, 2015.
Community received critical acclaim for its acting and writing, appeared on numerous critics' year-end "best-of" lists for 2009, 2010, 2011, 2012, 2014, and 2015, and developed a cult following.
ABOUT Entertainment section:
We are all about the gaming but we also want to bring the entertainment to people who might not really be big into games. This is why we created "Entertainment" category. In this category you will find pretty much everything besides gaming. So movies, animals, life and everything that surrounds us. Tons of quizes and we are pretty sure that everyone will find something for themselves. We are constantly working on new quizes and try to add them to site as often as possible so if you hit the bell in the right lower corner, you can always be up to date with newest quizes thanks to push notifications. We will not spam you so if you are quiz freak like us, consider subscribing. It is also a big signal for us that you like what we do! Without further a-do, check this quiz out, try to get the maximum answers and try out other quizes. If you have fun please share them with other people! Thanks and have a great time!


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Enjoy faster, easier checking with the use of PCMCU's Debit Card at no extra fee. Present your EMV-Chip debit card at the payment terminal to make a purchase. PCMCU's EMV-Chip cards make for a more secure way to pay! Apply for a PCMCU Debit Card now!
Debit Card Features:
- Debited directly out of checking account
- Accepted wherever Mastercard is accepted
- Your Debit Card can be used at all ATM Machines that accept Mastercard and Accel
- Extra safety – if your card is lost or stolen, you are not liable for unauthorized purchases
- The first 6 ATM withdrawals per month are free. Each additional withdrawal at an ATM not owned by PCMCU will cost $1.00
- Point of Sale (POS) transactions are always free!
- Free ATM access at any PCMCU-owned ATM Terminal
If your card is Lost/Stolen or for inquires about your PCMCU DEBIT card call (833) 202-2877
For a listing of no-fee ATM locations visit www.allianceone.coop
As a member of PCMCU, which is a member of the ATM networks, you can avoid surcharges at ATMs owned by other participating financial institutions. Members of all credit unions participating in this network, can look for the Alliance One logo on selective surcharging ATMs and know that they will not be surcharged even if:
• That financial institution ATM surcharges cardholders of other financial institutions; or
• There is a surcharge sign on the ATM that also displays the Alliance One decal
Here are PCMCU's Terminal Locations:
PCM Credit Union
601 Willard Drive
Green Bay, WI 54304
(Around the back of the building facing Ashland Ave.)
PCM Credit Union
1740 Stephenson Street
Marinette, WI 54143
(At the last lane of the drive up)
Paper Converting Machine Company
2300 S. Ashland Avenue
Green Bay, WI 54304
(Vending Area – for employees only)
CONNECT, powered by American Family Insurance
3500 Packerland Drive
De Pere, WI 54115
(Branch office – for employees only)
Robinson Metal
1740 Eisenhower Drive
De Pere, WI 54115
(Vending Area – for employees only)
Paper Converting Machine Company
660 Cormier Road
Green Bay, WI 54304
(Vending Area – for employees only)
Paper Converting Machine Company
1163 Glory Road
Green Bay, WI 54304
(Vending Area – for employees only)


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Reason and details of the disclaimer
You confirm you accept this disclaimer in full and failure to do so precludes you from using our website. Plan Your Holiday will be permitted to amend the rules relating to the use of our website and it is your responsibility to ensure you are familiar with the latest terms and conditions. You would be legally responsible even if you failed to read the terms and conditions.
Publication Rights
Travel Strong holds the copyright and intellectual property rights to all the content on our website and as such this prevents you from using it online or offline without our approval. This restriction applies to use on websites, social media and any other online system. Any content that you add to our website is legally owned by Plan Your Holiday and we have the legal right to change or remove your content if we choose to do so.
Refunds to Credit Cards
How do I obtain a receipt?
The Self Service section of our website is available for you to receive a PDF copy of the receipt for your travel arrangements made with us. You can also use the section on our website marked as "manage my bookings".
Does Plan Your Holiday mail hard-copy receipts?
No we only provide receipts in PDF form via our website as this helps to keep your travel costs down and ensure you have more money to spend locally in the resort when you arrive.
How long does it take to receive a refund?
If you are due a refund it will be processed straight away via our bank. It will normally take around 10 working days for the money to be credited to your account. Occasionally it can take as long as 15 to 30 days but this is not something we have control of. Your credit card provider will convert the funds into your local currency. The entry on your statement from the credit card company will show our company name of Plan Your Holiday against the transaction.
Is it possible to receive a receipt showing the name and address of my business?
Yes receipts for your company are available from the Plan Your Holiday Self Service and "manage my bookings" areas of our website. The receipt will be a PDF document that you can print and retain for your business records.
Is it allowed to pay with my friend's credit card?
Yes of course but we might need to check with you for additional security clearance when you book with a credit card belonging to someone else. This is standard practise and something we do on a regular basis for our clients.


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National Geographic will kick off their 9th annual SharkFest with "Shark Beach with Chris Hemsworth" on July 5th.
The hour-long documentary will have Chris diving with nurse sharks, meeting with experts and checking out the latest tech to prevent attacks.
Chris' home country of Australia recently had one of the worst shark attack seasons in a century. He wants to find out how people and sharks can co-exist better.
Source: Yahoo
Article Pic: Getty Images


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UFA-Dabei 836/1972 01.08.1972
01. German soccer Olympic selection - Finland 1:2 run players on course during training. Coach Jupp Derwall, great. Foot shooting ball, big. Goalkeeper keeps, Mr. Boerner interviewed Derwall, o-ton: "we consider ourselves as an outsider. What we want is to become group winners in our group, where Malaysia, Morocco and the United States plays, because we want the Russians out of the way, who will most likely want to group in the Group 2." Tracksuit training game. Players giving interviews, interview: "all Eastern bloc countries, all preceded Hungary and Russia."-"I would also say: Russia, Hungary, GDR, perhaps Poland still." - "I believe that any place between 5 and 6 would be a huge success for us." Match against Finland. Player misses on the target. Finland player shoots 1-0 into the empty goal. Viewers, children and police hat. Playback before Finnish goal. Ball goes out. Player falls on reporter who sits on the sidelines. Hustle before the Finnish goal. Penalty. Jürgen Kalb Award for Eintracht Frankfurt shoots compensation. 2. goal for Finland. Finnish players embrace.
02. Bremen University drive recording University of Bremen with various buildings. Newspaper headlines: Spartakus University of Bremen – APO University - University of the Red retort. "Is the University of Bremen as red?" Founding Director Dr. der Vring speaks o-ton: "you can say, we are on one side of the progress, we are geared for the reform on one side. The program that we have developed is also discussed by other universities, recorded, and I believe we will get through with these notions." Entrance hall of the University. Student, immediately before the University lecturer before the Board, an employee in the Office give interview about the University, o-ton: "In the University of Bremen the projects identify the contradictions of social reality, partially to learn and interpret. possibility to my opinion tends to" - "old universities have old University teacher, learned old things which are themselves hardly able to learn and new content compared to blocking behavior. In this respect allows the University of Bremen to make scientific access to topics that otherwise fall under the table in the discussion."-"I believe the practical benefits of the Bremen model for the service sector, it is so that the workflow is much more on collaboration and reporting relationships can be mined. Then added, that in the Committee's work it is the selected service providers possible, to work there himself creatively with the bureaucratic issues the University."-"the conception of this University is that we want to plan the curriculum of the University makes sense for the first time. The planning measure want to fulfil the requirements of the profession and from the beginning of the practice in the University to bring in." Session of the Chamber of labour. Discussion by members of the University with works councils from Bremen holdings. Cooperation between professors and students at the common points of the curriculum. Sign above the door calm and peace be granted to this House.
03. specialized boarding school for dog training cigarette box containing stars and dogs - dog boarding school of Antonienwald. Box will open. Owner of dog boarding contagious cigarette and speaks o-ton: "from the outset, hm, I went new ways... "Men with dogs on leashes are around him. Barking dogs in the kennel, great poodle is washed, gets cleaned teeth and is brushed. A spray. Kitchen. Dog food is ground through meat grinder and poured into a bucket. Boxer is trained to touch any other food. Sitting next to the table. Dog trainer looks in monitor, if dog food goes. Fierce noises. Dogs in restaurant refuse sausage that is offered. Walk through woods. Dogs are made safe by noise. Walk through traffic. Certificate and photo for hotel pass. Boxer welcomes wedelnd masters and mistresses. Young people and women give interview, why she gave the dog. Boxers at the wheel from moving cars.
Origin / Type of content
Persons in the Film
Boerner, Hans Jürgen ; Mouhuys van ; Vring von der ; Derwall, Jupp ; Kalb, Jürgen
Bremen ; Italy ; Antonienwald ; Munich
Sachindex Wochenschauen ; Higher education ; Dogs ; Interviews ; Football ; News, communications ; Smoking ; Sports details, fouls ; Dogs ; transport
Newsreel (G)
Weekly Newsreel
Translated by Microsoft Translator
- Title:
UFA-Dabei 836/1972
UFA-Wochenschau (Other title)
- Country of Origin:
- Federal Republic of Germany
Year of Production and/or Release
- Date of Release:
- 01.08.1972
- Year of Production:
- 1972
- Origin:
- German football Olympic team against Finland interview: coach camera: Rieck, Zaki, Ahsendorf Bremen University camera: Ahsendorf technical school for dog training camera: Ahsendorf at the beginning and end of total length
Technical Data
Have you seen these gems?
UFA-Dabei 988/1975
UFA-Dabei 663/1969
Welt im Bild 184/1956
Die Zeit unter der Lupe 1008/1969
Welt im Film 358/1952
Welt im Film 329/1951
Welt im Film 100/1947
Der erste Schritt
Welt im Bild 57/1953
Neue Deutsche Wochenschau 393/1957


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Online versus classroom training
Which is best for safety training – online or classroom?
Answered by Scott Wallace, production manager and authorized OSHA trainer, Summit Training Source Inc., Grand Rapids, MI.
Although we wish it were an easy "yes" or "no" answer, the fact of the matter is that one training format is not better than the other. The type of training you use is going to be specific to your needs and the topics on which you are training. The most important aspect is the quality of the training. According to adult learning theory, training must be engaging for the student and include:
- Clear learning objectives
- Content focused on the stated learning objectives
- Content tailored to the students
- Interaction between students and the training curriculum
- Application of knowledge in real or typical scenarios
To best determine the right training format, ask yourself these questions about your resources and your training population:
- What technology do I have available at my location? Do all of my sites have the same technology? Are the students familiar with computers?
- How many shifts do I have at this location? Is it feasible to bring employees into the classroom in groups? Does individual, self-paced training make scheduling around production time and shifts more efficient?
- Is my student population widespread? How important is the consistency of the training information?
- What is the students' current knowledge level of the subject?
- Do all of the students speak and understand the same language?
- Is this initial or refresher training? How often do I need to train on this subject? How many employees am I training at a time?
- Does my topic require hands-on training to meet compliance with regulations or standards?
Online training offers significant benefits in relation to cost savings, especially for large organizations or widespread employee groups where scheduling or travel can be cost prohibitive. However, online training that does not include interactions or multiple media designs to engage students in the learning is ineffective.
When online training is produced at top standards, it allows students to begin applying the skills they learn via interactions. Because the online learning environment allows students to begin applying knowledge during the course of learning, online training may result in noteworthy time savings.
Classroom training fosters more group involvement, team building and team problem-solving during the learning experience. Classroom training also allows for real-time instructor-to-student interaction. However, this may be prohibitive for students who are intimidated by asking questions or "saying the wrong thing" in front of their peers.
For many tasks in the safety industry, a class/laboratory approach works well. The class portion can take place in a classroom or online, with the lab section taking place in the field. Some professionals use a blend of online and classroom learning for initial and refresher training. No matter the format you use, engaging students in learning will offer the best results and return on your investment.
Post a comment to this article
Safety+Health welcomes comments that promote respectful dialogue. Please stay on topic. Comments that contain personal attacks, profanity or abusive language – or those aggressively promoting products or services – will be removed. We reserve the right to determine which comments violate our comment policy. (Anonymous comments are welcome; merely skip the "name" field in the comment box. An email address is required but will not be included with your comment.)


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What is the role of the Governing Board?
In the UK, school governors make up the largest volunteer workforce in the country! School governors have an incredibly important role. School Governing Bodies are given a range of duties by Parliament that concern the overall management of the school.
At King's Wood, under the leadership of the Chair (Mr N. Horslen) and Vice-Chair (Mrs A. Hickman), governors work in close partnership with the Head Teacher and school leaders to agree the aims of the school and to decide together how the standard of education can best be monitored and improved.
Our governors bring to this role a range of interests, experience and expertise, meeting as a full governing board once a term. They also serve on committees, to consider matters delegated to them by the full governing board. Any recommendations arising from such committee meetings are always referred back to the full governing board for final approval.
Minutes of meetings of the full governing board are displayed below once they have been approved. These are also available from the school.
When governors visit, they make every effort to observe the school in action and offer support to school leaders. Governors also try to attend as many extra-curricular school events as possible.
Governors normally hold office for four years from their date of appointment.
What is the role of Parent Governors?
Parents have a unique opportunity to become involved in the strategic direction of the school, by becoming parent governors. Please click here for further information about this important role.
If you would be interested in joining the Governing Board, please speak to the Head Teacher.
How can I get in touch with Governors?
All governors can be contacted via the school.
Governor Roles and Meeting Attendance
Please click here to see this information.
Full Governing Board Meeting Minutes


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Sponsored - World's largest ever crane auction2 March 2017
A two day public 'disposal auction' that includes 130 cranes (from 220 ton down to 35 ton All Terrain and City cranes) plus more than 500 significant lots of general construction equipment from the administration of Hewden Stuart Ltd will be sold by Euro Auctions at the Newark Showground in Nottinghamshire at the request of the administrators at Ernst & Young on Thursday 02 /Friday 03 March.
"This will be the single largest collection of used mobile cranes ever assembled for sale in one place, anywhere in world," stated Jonnie Keys, Operations Manager for Euro Auctions, Europe's largest auctioneers of industrial plant, construction machinery and agricultural equipment. "Phenomenal interest is being generated by this auction within the industry from right around the globe as all the cranes have been maintained in A1 condition and many have exceptionally low hours and come with some manufacturers' warrantee."
The 130 cranes are all scheduled to go under the hammer at the Newark auction on Friday 03 March with the sale of over 500 lots of low hours construction equipment and machinery from the Hewden Portchester and Dagenham depots taking place the day before
Register and view all equipment on www.euroauctions.com


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Casio Lovers Collection-limited edition of matching watches for couples
The Lover's Collection of matching watches for couples features a G-Shock for him and a Baby-G for her. The LOV-09A and LOV-09B sets for 2009 include models with combined digital and analog displays, as well as digital-only models. The designs and colors for these paired G-Shock and Baby-G models are perfectly matched.
The back of each watch is engraved with the year 2009 along with the good and mischievous cupids, the design motif of the Lover's Collection. This motif represents the eternal attraction between opposites, and symbolizes the relationship between lovers. Both sets are ideal for lovers that wish to celebrate their special moments together with matching timepieces.


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Friday, August 3, 2018
Did you know that hundreds of thousands of people in the United States commute to work by bicycle, according to the U.S. Census Bureau? That's a lot of cars off the road each day, a lot of gasoline saved and a ton of...
Crescenta Valley Insurance is an independent insurance agency located in La Crescenta, California.
Give us a call, stop by, or request a quote online to find out how much we can save you on your insurance.


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By Pranshuta Arora (Team CB)
Hello beautiful ladies! I'm back with the featured closet and the closet which has caught my attention this fortnight belongs to the gorgeous Shubhi Garg. Her collection of dresses just took my breath away and with winters almost there, there can't be a better option than these stunning dresses to enjoy the chilly weather. So have a look at some of the top picks from her closet.
I'm assuming that all you ladies have already downloaded the app by now. If still something has kept you busy and if you have not done it yet, do it now.
Her love for dresses is quite evident from her closet. You will find beautiful collection of dresses which are quite versatile in their style.
You can get your hands on this beautiful green dress for just Rs.1899. Team it up with gumboots or even ankle length boots.
This pure cotton khadi dress is the most perfect ensemble for winters. The checks will grab all the attention and the stylish collar will add to the style quotient. You can get it from the app for just Rs.999.
This gorgeous white dress is by the label of Stalk Buy Love. She is offering it at a price of just Rs. 499. The net makes it look elegant and delicate. It can be worn for a light afternoon.
The cute multi coloured dress is up for sale at Rs. 2499. You may sport your really high heels with it or you may also wear boots and a leather jacket with it to make it look bolder.
Even her tops and shirts collection is quite commendable.
The beautiful multi coloured top which can also be worn as a dress is up for sale at Rs. 1899 on the app.
I'm so sure you will fall in love with this Indo Western pink top. It would go with white denims, leggings or even white palazzo pants. You can pick it up from the app for just Rs. 1499.
White is my favourite and so it's quite obvious that I just love this white top from Globus. But even you ladies would love to make this top yours forever. It has been priced at just Rs. 699.
A pair of denims is something we all need and no matter how many pairs of denims we have it's never enough. Adding this stunning pair of denims from Forever 21 at a price of justs.899 would be a wonderful idea.
What an adorable jumpsuit isn't it? The Stalk Buy Love beige jumpsuit is being offered at Rs. 699 which is nothing for this smart jumpsuit. Shubhi has teamed it up with a black leather jacket here which looks amazing.
Her bags are equally good. This Stalk Buy Love orange bag can be bought for Rs.899 from the app and I think it's totally worth it.
So go for it girls. Pick up your favorite and stay tuned for the next featured closet. You can also write in to us if you want us to feature your closet.
Just in case you live in the city of Nawab's Lucknow, you may borrow the entire ensemble( outfit+ shoes+ bag+ sunglasses +jewelry) for just Rs.499.


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The Loop
Dove Mountain's 15th hole the spot to be at the Match Play
MARANA, Ariz. -- Drivable par 4s are a staple of many PGA Tour courses. They're fun for fans to sit and watch as players are challenged as to how aggressive they want to be off the tee.
The 15th hole at Dove Mountain G.C. fits that bill and then some when you add the strategic nuances that come with match play. It was arguably the most interesting spot to watch the action Wednesday during the first round of the WGC-Accenture Match Play Championship, with a large hospitality tent to the right of the tee -- and all the accompanying distractions a hospitality tent offers -- providing as much entertainment as the action on the course.
Here were some of the highlights:
-- Rory McIlroy and Boo Weekley had almost a half hour wait on the tee as in front of them Brandt Snedeker and Webb Simpson both found the cacti off the tee. During the wait, the two got to talking about tennis, Weekley curious how good McIlroy had become at the sport considering his fiancee, is one of the world's best female players. "I practiced about a half hour every day when I was home in the offseason," McIlroy said. Who thought a country guy from northern Florida and a kid from Northern Ireland would bond over tennis?
-- Weekley, after chatting a bit with the fans in the hospitality tent, pleased many by grabbing his driver. Hearing their approval, he deadpanned: "Well shoot, I ain't gonna get it up there on the green even if I picked it up and threw it." When his tee shot just cleared the fairway bunker, Weekley gave a big fist pump to the cheering crowd.
McIlroy edged Weekley, 3 and 2, during their first round match at Dove Mountain. Photo: Ted S. Warren/AP
-- The 15th-hole captain, Dave Johnson, had his hands full with the rowdy fans in the tent. "I have at least four cell phones a group," Johnson said. "And they're sneaky about it. You yell at 'em and then they'll hide the thing and take it out again when you turn around."
The 78-year-old Johnson has been the hole captain for four years on No. 15, which had five marshals around the tee, and a volunteer at PGA Tour events in Tucson for 17 years. While there are some downsides to working the hole, he says he enjoys it: "It's where all the action is."
-- Aside from the incessant cell-phone photo snapping, there were loud whistles and spilled beer cans along the 15th tee. It wasn't quite the craziness you'll find when you go up I-10 to the 16th hole at TPC Scottsdale, but they're in the same ballpark. McIlroy, 3 up in the match at the time, couldn't help but laugh as he observed it all while standing on the tee.
-- USGA rules expert Bernie Loehr came to Webb Simpson's assistance to help assess whether he could move some rocks around his ball. Some jumping cholla got stuck to Loehr's foot, so Simpson's caddie, Paul Tesori, used Simpson's wedge to try to scrape the thorns off his pants. Ten minutes later, Loehr still had some thorns stuck in there. "Good thing the cholla didn't jump too much," said Loehr.
-- Ahead in his match, McIlroy was still aggressive off the tee. His drive, however, sailed left of the green, hitting a spectator who in an attempt to dodge the incoming ball had fallen back into a cactus bush. In turn, the fan had the jumping cholla on his back as a souvenir. McIlroy would lose the hole, but eventually win the match, 3 and 2. No word though whether he, after famously giving lodging and tickets to a fan at the British Open whom he hit with an errant shot, offered the fan at Dove Mountain a souvenir to go along with the thorns.
He joked he wanted to see who Butch would be rooting for in this case. "


text: The majority of my sculpture is concept driven and is often highly viewer interactive. The conceptual component of my work is often derived from a memory that is stirred by the shape of some found object or from some memento that I have held onto since childhood. These things bring up thoughts and experiences that challenge me and guide me through the creative process. There are elements in much of my larger works that are derived from my love of fantasy, escapism, and pop-culture imagery. I find that it is not always necessary to me that the viewer understand my concept, but it is important to me that I provide a visually exciting experience that might encourage the viewer to see and engage with art more often.


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The following clients are now live with their online stores, powered by Nexternal Solutions. Java Frost. Blending it's way to success, Java Frost has become a leader in the blended and hot beverage industry. They sell a variety of gourmet drink blends and gourmet coffees. Visit their online store at www.javafrost.com Cincy Favorites love Cincinnati […]more
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