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Lecture 12: Discrete vs. Continuous, the Uniform | Statistics 110
So, the main topic for the next couple
lectures is continuous distributions. We've learned about the binomial,
and the poisson, and the hypergeometric, and so on, and
at this point we've covered all of the famous discreet distributions
that we need in this course. And no now is a good time to start talking
about the continuous distributions. I like to do discreet before continuous, because conceptually it's
simpler to think about discreet. But it doesn't mean that
continuous is harder, necessarily, because discreet is kind of
conceptually easier in a sense. But, on the other hand, we have all
these nasty sums that come up, and so we learn some ways to sometimes
avoid the sums using stories, and so on, but sometimes you just have
the sum you can't deal with. The continuous case, we'll be doing
integrals instead of sums, and even though this sounds counterintuitive, in general,
it's easier to do an integral than a sum. Although the same thing could come up, we could be faced with integrals
we don't know how to do. So again, we're gonna try to look for
kind of more clever, and more conceptual ways to avoid having
to do lots and lots of integration. But anyway, we'll come to that later. But, a lot of the ideas
are completely analogous. So, at this point, I'm assuming you
have a pretty good understanding of what a PMF is, and
what is a discreet distribution? What does it really mean, and the expected
value of a discreet distribution, and now we're just gonna move
into the continuous case. So, I think, and just for
having a big picture on this, it helps to just kind of
contrast the two things. So I'm gonna make kind of a dictionary
of discrete world and continuous worlds. So we can put discrete world over here and
continuous world over here. So we have a random variable
that we're looking at, and usually we've been calling our random
variable x in the discrete case, and usually we'll call it
x in the continuous case. So, so far it's completely analogous. We got discrete, continuous. Now in the discrete case, as you're very
familiar with by now, we have a PMF, Which you can just think of as the P(X=x),
viewed as a function of little x. So if it takes positive integer values,
then I would need to specify this for all positive integers x. In the continuous case, the [P(X=x)=0]. So in that case we have a PDF instead,
which usually we would write as f(x), but
you can call it whatever you want. I'll call it f sub x (x) just to
emphasize that this is the PDF of x. So I'm gonna tell you what a PDF is,
but I'm just telling you now, that it's analogous to a PMF. The reason we need this
is that the [P(X=x)=0]. So continuous, it means we're thinking of
random variables that could take on any real value, or
maybe any real number in some interval. So say we had the interval
from zero to one, and X is allowed to take on real
number value between zero and one. Well, I mean we could make up examples
where this not true, but in the continuous case every, there are uncountably many
real number between zero and one, and any specific number like Pi
over four has probability zero. So if we just try to write down a PMF
we would just say it's zero, and that would be useless. So that's why we need
something else instead. So, I'll tell you what a PDF is,
but that's the analogy. Just to continue this a little more, then I'll start telling you
more about what PDFs are. We have a CDF. That's this function F(x)=P(X=x), and sometimes we'll subscript the x just
because maybe if we add another random variable y, we could write F sub y for
its CDF, okay? Well, in the continuous case we
have a CDF, exactly the same thing. So that's one advantage, and
we've seen in the discrete case, usually it's easier to deal
with the PMF than with the CDF. The CDF in the discrete case is a lot like
a step function with all these jumps. It's not so easy to deal with,
and this is much more direct. But one virtue of CDFs is
a CDF is completely general. So every random variable has a CDF, and so
we don't need to separate out the theory. Now, let's talk about PDFs. So, now this is a PMF. So the PDF is the most common way to
specify a continuous distribution. PDF stands for
probability density function, not portable document format,
probability density function. Okay, so the keyword here is density. The common mistake with PDFs is to
think that they're probabilities. It's not a probability,
it's a probability density. So you can think of density,
just in an intuitive sense as like, think of probability as mass. Remember if the pebbles with
the total mass equals one? But in the continuous case we can't think
of pebbles any more, it's more like we just have this kind of massive of mud
that we're smearing around the space. So I think of discrete as pebbles,
continuous as mud. The total mass of the mud is one,
and density, makes you think of mass per volume, mass per area,
things like that, mass per length. Okay, so it's probability per something,
but not probability. So we say that x, this is a definition. A random variable, X, has PDF F(x) if in order to find probabilities for X we can achieve that by integrating the PDF. So, if the probability of
let's say X is between a and b, that is X is in some interval a b, must be given by the integral
from a to b of f(x)dx. For all a and b. So, f(x) is not a probability, it's
what you integrate to get probability. Integrated density,
then you get a probability. So that's the definition, and let's see
how this relates to CDF and other things. Notice, by the way, that if we let a=b, Then we're integrating from a to a,
of f(x)dx. So that's the area under the curve
from a to a, which is zero, cuz you haven't actually specified
any interval, so that's zero. Which agrees with what I said there, the
probability of any specific point is zero. We need an interval of non-zero length. Okay, so that's called a PDF,
and to be valid, remember, for a PMF, I said that a PMF is valid
if they're non-negative and they sum up to one, right? So by analogy, for
a PDF we want them to be non-negative, and rather than summing to one,
it should integrate to one. Okay, so to be valid,
f(x) is greater than or equal to 0, and the integral of f(x) for
minus infinity to infinity should equal 1. Otherwise, we have not
specified a valid PDF. So it might look something like, just to draw a picture,
an example, maybe a PDF. A famous example would be the bell curve
type of thing that we'll get to later. But anyway, for
the purpose of the picture, I don't really care exactly what
the definition of this function is. But I'm just drawing some curve
from minus infinity to infinity. Now it might be that it's
0 on the negative side and only positive to the right or whatever,
but it's some continuous looking curve. And the total area, if I shaded
the whole area under this curve, I would get 1, right? And so larger points where the density,
I drew a symmetric one, but it doesn't have to be symmetric,
it could be some nasty looking curve. As long as it's non-negative and
the area under the curve is 1, those are the requirements. So to kind of interpret a little more
of what does the density really mean? Cuz I said it's not a probability. If we take f(x), let's say, at some
point x0, what is that really like? If we take some point x0 here and
we say the density is this number. What does that mean? It's possible that this
number is greater than 1, for example, because you can have a function
that sometimes is greater than 1, but the integral could still be 1, right? So we can't say that's a probability, but
what we can say is, so this is a density. So if you think of it as like
probability per unit of length, then if we multiply by some small number,
let's say, epsilon is approximately
the probability that x will fall in an interval
of length epsilon. Let's call this, let's say,
x0- epsilon/2, X0 + epsilon/2. So all I did was take x0. The probability of the random variable
exactly equaling x0 is 0, okay? But if we take some tiny,
so for epsilon, very small. So the probability is 0 of it equaling x0,
but we take some tiny little interval around x0, I just wrote
down an interval of length epsilon. Then the probability is approximately the
density times the length of that interval. But by multiplying by this epsilon here,
we're kind of converting it back into a probability scale instead
of a density scale. And to see, so this is kind of a good
intuitive way to think of a density. But I haven't yet shown you why is that equivalent to this
mathematical thing that I wrote here. But to see why this is true,
just by staring at this, If we wanna find the probability,
then what do we do? We integrate the pdf from here to here,
right? So imagine this integral where you're
integrating from here to here, okay? And then let's think about
what would that integral be. Well, I didn't just say epsilon was small,
I said epsilon is very small. I could have said very, very small. Now if epsilon is very,
very, very small, what that means is that in that tiny little
interval, f is not gonna change very much. So over that tiny miniscule interval, we can treat this function as
being approximately a constant. And it's easy to integrate a constant, the integral of a constant is just the
constant times the length of the interval. And that's all we did, so we're treating
it, if it's approximately this constant on that interval times
the length of the interval. So, that's why this follows from this. And this is more useful for driving
things, but this gives you some more intuition on what's the difference between
a probability density and a probability. Okay, so, let's see how is
this thing related to the CDF? So if x has PDF, little f, let's find the CDF. Well, by definition, the CDF is
the probability, That x is less than or equal to little x, but by definition,
I said the definition of a PDF is that's the thing that you integrate
to get probability, right? So if I wanna know what's
the probability that x is in any region, all I do is integrate
the PDF over that region. So here, it would simply integrate
from minus infinity to x of, I could call it f(x)dx, but it's a little
bit clearer to change the letter, so f(t)dt. t is just a dummy variable here. I just didn't want it
to clash with this x. That is, for any particular number x,
we're gonna take this curve. Let's say x is here. If this x is this x we're looking at, then we're saying just look at the area
under the curve up to this point. That would give us the CDF at that point,
all right? Because we wanna know the probability
of everything to the left, and probability is given just by
taking area under this curve. So it's just the area under the curve
from minus infinity up to x. That's all we're doing. So that shows how to get
from a PDF to a CDF, okay? Well, what about the other way around,
if we have a CDF, how do we get the PDF? So go the other way around,
if x has CDF, capital F, and of course, we're assuming it's a continuous
random variable, not a discrete one. So in the continuous case, by the way,
the terminology, it could be slightly confusing because when we say we
have a continuous distribution, it means capital F should be continuous. But we don't just want
it to be continuous, we want it to be differentiable. So the continuous refers to not so
much to F being a continuous function. It refers to the fact that x can
take on a whole continuum of values, rather than just discrete values, okay? So if it has CDF F, and
x is a continuous random variable, and then we want to get,
From the CDF to the PDF, So f(x)=, so let's think about that. This is the relationship between a CDF and
a PDF, okay? And but now I wanna say, if we know this
integral, how can we extract out this? Well, the answer is just take
the derivative, right, f(x)=F'(x). And why is that true? Your favorite theorem of calculus, that's
the fundamental theorem of calculus, FTC. Actually, we're gonna need both parts of
the fundamental theorem of calculus, so it's nice that actually that
it is pretty fundamental. At least the way I learned it, part one of
the fundamental theorem of calculus said if you have an integral that looks like
this, up to some indetermined upper limit, if you take the derivative of that,
then you just get this function. So that's the first part of
the fundamental theorem of calculus. The second part of the fundamental
theorem of calculus says that if you wanna do a definite integral, you find anti-derivative and
then evaluate it at the two end points. So okay, this is just saying
the derivative of the CDF is the PDF, in the continuous case. So it's a very straightforward
relationship between them. And if we wanted to know, this also kind
of confirms something we did earlier. Let's say we wanna know the probability
that x is between a and b. And in the discrete case it's crucial
whether less than or equal, and so on. In the continuous case, it makes no
difference if you write strict or not strict here. So according to the definition of a PDF, if we wanna get the probability of this
interval that x is in that interval. All we do is integrate
the PDF from there to there. But another way to think about this would
be, remember your fundamental theorem of calculus, and
the notation matches up pretty well too. Because, like in AP calculus, usually if
you have a function little f, usually it will call its anti-derivative capital F,
which is exactly what we're doing here. If we wanna do this integral,
we take some anti-derivative. Well, we already have one, that's the CDF. And then we evaluate here, evaluate there. So that's just F(b)- F(a). So that's also true by
fundamental theorem of calculus. And that's similar to a result
that we had earlier for CDFs, so it's consistent with earlier stuff, okay? So we'll do some examples
in a little while. But right now this is just the general
framework, and making the analogy, okay? So we have a CDF, and I'll just add here,
just to have it in this dictionary too. That's the derivative of the PDF is
the derivative of the CDF, question?>>[INAUDIBLE]
>>Yeah, and the question is, in this framework,
is big F always differentiable? Yeah, we have to assume
that it's differentiable. I mean, there are functions
that are continuous, but not differentiable everywhere. But in that case it would just
be a more complicated thing. And when we say continuous random variable
in this course it means we have a CDF which has a derivative. Because if we don't have a PDF then we're
not dealing with continuous distributions and things can be much nastier. So yeah we're assuming that
this derivative exists. Okay, so that's, So
in general if I ask you, find the distribution of whatever,
in the continuous case, in the discrete case,
you can either give the PMF or the CDF. Those are equally valid ways
to describe a distribution. In a continuous case, you can give the PDF
or the CDF, those are equally valid ways. Okay, so let's continue this list. In the discrete case,
we have the expected value, right? And remember the expected value,
we just take the sum of the values times the probability
of the values, okay? So in the continuous case, this would just be 0 because all of
these are 0, so that's no useful. But by analogy, instead of a sum,
we'll do an integral. S the definition of the expected
value in the continuous case is that we integrate x times the PDF. So it's completely analogous. In general we're gonna integrate
from minus infinity to infinity. And sometimes we'll deal with random
variables where the only possible values are say between 0 and 1. And in that case we're just integrating
0 outside of that interval. So then we would restrict it to
the region where it's non-zero. But in general that's best the definition,
okay? So that's completely analogous. Let's do one more concept that applies in
both the discrete and continuous cases. And that's the notion of variance, so
we've been talking about expected values. But that's just giving a one number
summary of the average, right? But it is not telling us anything about
the spread of the distribution, right? How spread out is it? So for that, we need the idea of variance,
and the definition of the variance. So intuitively, variance is just supposed to be a measure
of how spread out the distribution is. That is, on average,
how far is x from its mean? So we might start by trying to do
the expected value of x minus the expected value of x. Here's the mean. This is the difference between x and
its mean. But if we just did this though,
we would always get 0, though. Because by linearity,
that's E of X- E of E of X. But E of E of X isn't E of X. Because it's just a constant, so this would be useful because
that would just be zero. Okay, so then I guess the most
obvious thing to me to do to fix that problem is to
put absolute value signs. Because then we're making
it non-negative and then there won't be 0 anymore,
except if x is a constant. But absolute values
are annoying to deal with. For example, the absolute value function,
it's this V shaped thing right? It has a sharp corner,
it's not differentiable. It is difficult to work with. So the standard way to deal with this is
instead of absolute values, to square it. One reason as I said is
that the absolute value is just annoying because
it's not differential. Kind of a deeper reason though is that
the square, anytime you see squares, it should start to reminding you
of the Pythagorean theorem, right? It means that there's a lot of geometry, there's a lot of beautiful geometry that
goes on with squares, and sums of squares, and right triangles, and
Euclidean distance, and things like that. And you lose that geometry if
you're using absolute value, and there are other reasons as well. But anyway this is the standard
definition of variance. So this is on average, how far is x from
its mean, except that we're squaring it. One annoying thing about squaring it
though, is that we changed the units. So if x is like a measurement that
let's say it's measured in miles. We are measuring some distance
in miles and we square it, we've got miles squared, okay? And so that's no longer in the same
units as what we started with. So because of that, something more
interpretable is the standard deviation, Which is a familiar term. Standard deviation is defined as just
the square root of the variance. So this seems, at first, like a kind
of convoluted thing to be doing. First we square everything then we
take the average then we square root it back again. The reason is that the variances
has really nice properties, but on the other hand we changed the units,
so we just change it back at the end. So that's the definition
of standard deviation. In general, variance is a lot nicer to
work with than standard deviation as far as doing the math. But then at the end of the day when you
want to have something interpretable. It's easier to think about what
the standard deviation means, because you're back on the original units. Okay, and let's just write one. One nice thing about this letter E
notation, this is a really good notation. E for expectation. Because I could just write
down this one thing and I didn't assume here that X is
continuous or discrete or anything. This is just a general definition and
I didn't need to write a separate definition for
the discrete or for the continuous case. So this is a unified definition. Let's just write the other. Another way to compute variance,
rather than doing this, so another way to write variance which
is more commonly used than this one. This one's the usual definition but the
other way to write it which I'm about to show you is usually easier for
computing it. Not always, sometimes this one's easier. Another way to express variance. So we want the variance of X. Let's just expand this thing out. I'm just gonna multiply it out, right. So that's X squared. -2 X(EX) I'm just squaring this thing + (EX) squared. And let's use linearity this
is E(X) squared, minus. Now for this middle term, the 2 is
a constant and constants can come out. The E(X) is also a constant, right? X is a random variable,
E(X) is just a number. The 2 E(X) is just
a number that comes out. So that's 2 E(X), and
then what's left inside is still an E(X). So we have another E(X) there, and
then plus, this thing is also a constant. So taking its expected value does nothing
because it's just a constant already. So that's + E(X) squared, and so this whole thing just becomes
E(X) squared- E(X) squared. And it sounds like what I just said was 0,
but the parentheses are different. Here we square it first
then take the average. Here we take the average then square it. We take that difference, okay? So that's usually easier. And so that answers the age old question,
if you had, this question came up for me I think in seventh grade
science class where I had to, do a bunch of experiments and
then I got a bunch of numbers. And for some reason I was squaring
the numbers and I wanted the average. I didn't know whether I should
square first and then average or average first and then square. And I think I computed both ways and
I got a slightly different answer. Which one is correct? Well, this one doesn't say
which one is correct, but this says that this one will always
be bigger than or equal to this one. And equality holds only in
the case when X is a constant. So if X is a constant, then the variance
is 0 cuz X just equals its mean obviously. If X is not a constant, than what's
gonna happen is that this thing, that you're averaging some
numbers that may be sometimes 0. But it's certainly sometimes positive,
and you can't average positive things and get a negative number. You can't average positive things and
get 0. So it would be strictly positive. Which means this is
strictly greater than this. Except in the case of a constant. So, okay, that's the variance. And as far as notation,
it's standard to write E(X) squared for E(X) squared this way. That's just standard notation. So, if you see E(X) squared, you should
always interpret that as squaring first, and then take the E. That's just a convention,
a pretty standard convention. This way is a little more clearer,
to avoid any possible ambiguity. But, it's very common to
see it written this way. So interpret it as squaring first. Okay so that's variance, and over here
we can continue our little dictionary, variance of X=E (X squared)-
E (X) squared the other way, And then the continuous case,
same thing again. And the one difficulty with this is,
we've been talking on how do we compute E(X), but
how do we actually compute E(X) squared, that's the question that
we need to address. How do you actually compute that thing? So we'll talk about that a little later. But first, we should see at least one
example of a continuous distribution. The simplest one to start
with is called the uniform. As far as what you'll
need before the midterm, there are only two continuous
distributions that you need to know by name before the midterm,
and then we'll do more later. One is the uniform,
the other is the normal. Uniform is the simplest
continuous distribution, so we'll start with that one right now. Normal distribution we'll talk about
mostly for next week, and normal distribution is the most famous important
distribution in all of statistics. And the reasons why it's so important will kind of gradually
emerge over the semester. Let's start with the uniform. So here's the uniform distribution
on some interval on (a,b). So we have some interval from a to b. I'll say here's a, here's b. We wanna pick a random
point in this interval. I'll put random in quotes. In this interval. How do we do that,
the question is what does random mean? If it's sort of intuitively
random is too vague because that just means we have some
random variable okay? What if we said completely random. Like what's the most
random that it could be? Again that's a little bit vague but let's just kinda explore that intuition a
little bit and then write down a formula. If it's completely random see I can
just see the probability of any two points is the same because all real
numbers between here and here. Every individual number is probably 0. So it's not so interesting to say
all the probabilities are the same. So pick some random point say,
there, x but the problem if I got that
exact value right there was 0. Okay so that means we still have
the same way it does mean for it to be completely random. So well the intuitions
now is suppose we broke this interval into two halves
where this is the midpoint say. Intuitively, if it's completely
random it should be that this half is equally likely as this half. Cuz If it were not then it seems like we
would kind of prefer to be, you know, the random variable prefers to be
more to the right than to the left. And somehow we want a concept
where it's not gonna, it doesn't care where it is, right? So in other words we could
say that probability, so for the uniform means that probability
is proportional to length. That's a reasonable definition. That is, if we take two
intervals of the same length, they should have the same probability. If one interval is twice as long, it seems reasonable that that
one should be twice as likely. So we're just gonna write down
a continuous distribution where probability is
proportional to length. And so to specify this, we can
either write down the pdf and drive, the cdf, or we can try to figure out what
the cdf should be and derive the pdf. Let's start with the pdf here,
because we're trying to practice pdfs. So here's the pdf, it's a constant. If x is between a and
b and it's 0 otherwise, Because I want probability to
be 0 outside of this interval. Inside that interval I want the density
to be constant, because if the density were higher at one point than another,
that doesn't seem very uniform. So well of course, we could ask, what's c? Well it has to be that
the integral of the pdf is 1. And I could start out by integrating
from minus infinity to infinity, but of course we only need to integrate from
a to b, cuz it's zero outside of there. If we integrate this we have to get 1, therefore c equals1 over b minus a. So it's just one over
the length of the interval. It has to be this way otherwise
this would not be a valid pdf. Now suppose we want the cdf. So to get the cdf we just have to integrate this thing
minus infinity up to x. So how do we do that? Again, we don't really have to go
all the way from minus infinity we can just start it at a. f of t dt,
then we have to consider some cases. Well, first of all this
is 0 if x is less than a. Well, this expression here that I wrote
down is kind of already assuming that x is greater than a. So assume x is greater than a. If x is less than a, then the probability
is 0 so the cdf has to be 0. And we also know that it's 1 if well,
let me just write this separately. So here's the cdf,
if x is less than a it's 0. If x is greater than b it's 1, because
we know for sure that x is less than b. Now, the interesting case is
what happens in the middle. To get the thing in the middle, all we
have to do is integrate a constant and this is the constant with the integral so
that I plug in f of t equals c here. That's gonna be c times x minus a. Right just integrate the constant. It's a very easy integral. So that's just gonna be x
minus a over b minus a. If x is between a and b. And notice this makes sense because if
we let x equals to a here it reduces down to 0. And if we let x equal
to b it reduces to 1. So this is a continuous function. So it's saying intuitively,
this is a linear function of x. They're saying that the probability is,
as you increase x, the probability is increasing linearly. Which make sense, cuz you're
accumulating more and more stuff. So let's get the expected value of x. Expected value of x, again,
it's just gonna be an easy integral. Because we just have to
integrate from a to b of x times the pdf. So I just wrote down x times the pdf. So integrating x is easy,
it's just gonna be x squared over 2. So this is x squared
over 2 times b minus a. And we evaluate this
as x goes from a to b. So that's really just b squared. Let's factor our the 1
over two 2 b minus a. And then it's b squared minus a squared. But b squared minus a squared
is b minus a, b plus a. So we can actually cancel the b minus
a and we just get a plus b over 2. Just doing that easy integral. Well, that's a very intuitive answer. That's just the mid point. It says the average is in middle which
it would really be weird if that didn't happen cuz this is supposed to be uniform. Okay, so that was just check that. Now, we have a bit of a quandary though. For how to deal with the variants. So let's try to get the variants. So If we want the variants then that
means we need e of x squared. Because we know this part
we don't know this part. How do we get rid of x squared? Well, E of x squared, Equals? So if we think carefully about
this how do we get E of X squared? Well, X squared is a random variable. Let's call that thing Y. So let's let Y equal X squared. If we take a function of a random
variable it's a random variable. So Y equals X squared. So that's E of Y. And how do we get E of Y? Well to get E of Y then
we need to know the pdf, assuming X is continuous right now. To get E of Y then we need
to know the pdf of Y and then we integrate Y times the pdf of Y,
it'd be Y. So the question is do
we need the pdf of Y? But that sounds kind of annoying
because we don't know the pdf of Y. Now we can get the pdf of Y, and later in
the course we will talk about how do we get the pdf of Y, but right now that's
seems like a pretty annoying problem. So let's kind of do this
more carelessly instead. Let's just say well it's too
much hustle to get the pdf of Y. So instead I'm just gonna say
I'm gonna reason by analogy. And I'm looking at this
formula right now for E of X. But I don't want E of X. I want E of X squared. So I'm just gonna change
that to an X squared. All right, I want X squared, not X, so
I'm just gonna put down X squared there. And then I'll go f of x dx. That's the pdf of X, that's what I know. And I'm too lazy to find the pdf of Y,
so I'll just change X to X squared. Well, that doesn't sound very legitimate. This, what I just did is called the Law
of the Unconscious Statistician. Which has a nice acronym
that's just LOTUS. It's called that because that just
seems like if you're kind of like half asleep and
you just want to find this thing and you just kind of replace X by X
squared because X squared and it seems like something you might do
if you're not thinking very hard. So to state it in general
in the continuous case, we want the expected value
of a function of that. X is a random variable who's PDF we know. We want the expected
value of a function of X. So, the Principled Approach would be, find the distribution of this and
then work with that. The Lazy Approach would be,
still use the distribution of X but that sounds kind of too good to be true. So the Lazy Approach here would be well, I'm gonna take g of X I am
gonna change big X to little x. And then I am still gonna need
insist on using the density of x and not convert anything. Well, this turns out to be true. So I'll put a box around it. We can talk sometime next week
about the proof, why this is true. But this turns out to be true. And thus, even though it sounds too
good to be true, it actually is true. So that's called LOTUS. This is the continuous version. In the discrete let me
write both versions. So a continuous LOTUS is
that thing I just wrote, we have LOTUS so
same equation you can copy that there. And let me just write the discrete case, again we want the expect value
of some function g of S, so all I'm gonna do is take this. This is the definition
of the expected value. All I'm gonna do is change X to g of x. So this is gonna be g of
x times the PMF of x. It says we don't need to convert and
get a distribution for g of x. We just do that. This is also valid. We'll talk more about why later. But it's useful to know that right now. So coming back to this
problem about the uniform, if we want the variance of the uniform so let's let, just for simplicity let's
let u be uniform between 0 and 1. And suppose we want the variance. So we know the expected value of u,
Is one-half, just the midpoint. And if we want E of u squared. According to LOTUS, we don't need
to first find the PDF of u squared. We can just directly write down
the integral 0 to 1 of u squared times the PDF times the PDF f
sub u of u du as the PDF, but this PDF is actually equal to a constant
and that constant is 1 in this case. So this is just equal to,
this part is just one. So it's the integral of u squared, the u, u cubed over 3, which is 1/3. So therefore the variance
of u equals e of u squared, minus e of u squared the other way. And that's one-third minus
one-quarter equals one twelfth. So the variants of a uniform
zero one is a one twelfth and that was a very easy calculation
because we were able to use lotus here, which we haven't proven yet but
we will talk more about that later. I'm showing you how to use it right now,
then we'll justify it more. So that thing that's too good
to be true actually works. So that's Lotus. One more thing about
the uniform distribution. It seems like the uniform is the simplest continuous distribution that
you could possibly imagine. Because the PDF it's just a constant. On some interval and one other point about this is we have
to have some bounded interval here. We cannot define a uniform
distribution on the entire real line. Sometimes that it's a bit
annoying if there isn't one. But if the whole real line there
will be no way to normalize it, there'd be no way to find a constant and
make it disintegrate to one. So it sounds like this is
an extremely simple distribution. And it is,
it's just constant PDF on some interval. Extremely easy. So start with the uniform zero one,
it seems very simple, but actually uniform zero one
has the property that if you give me one uniform random variable
and you're interested in some other distribution, there is a way to
convert it and simulate that. That is from the uniformed
zero one you can simulate or generate from any distribution
no matter how complicated it is. At least in principle. As a matter of computation that may be
easy or hard, but in principle from the uniform you can get anything, so
I call that universality of the uniform. Universality of the uniform means
that given a uniform you can create any distribution that you want. So that's kind of theoretically nice
in that it kind of unifies concepts and says this things that's seems very,
very simple to just one uniform. You can actually use it to generate
something that's as complicated as you want. That's kind of cool but it's also useful
in practice where most computers programs can generate random numbers between zero
and one, it's actually pseudo random. But they not know how to generate
whatever complicated distribution you're interested in. And this, in many cases gives you away to convert from the random uniforms
to whatever you want to simulate. So I want to show why that's true. So the statement is that,
we're gonna start with the uniform between 0 and 1 and let F be a CDF, that we're interested in. So usually we've been talking about here's
the random variable and then find in CDF. Here we're going the other way
in the sense that we assume that we have some CDF
that's of interest to us. But we do not yet have access to
a random variable that has that CDF. So let F be a CDF and
it's possible to generalize this further. But to make this something that we can
do fairly quickly let's assume that F is strictly increasing, so
we don't have to deal with flat regions. And let's also assume that F
is continuous as a function. Just so that we don't have to
think about jumps right now, although you can generalize this. Now then the theorem says that if we let X define X to be F inverse of u. So the inverse function exists in this
case because I took something that was continuously and strictly increasing,
it will have an inverse. So we take the inverse and we plug in u. Then the statement is that X
is distributed according to F. That is the CDF of X is F. So what this says is we have
this CDF we're interested in. We take the inverse CDF, plug in
the uniform, and then we've constructed a random draw from that distribution
we're interested in, capital F. So let's prove this very quickly. And the proof doesn't require
anything fancy at all. It doesn't require anything, except for
understanding what a CDF is. So another reason I like to talk about
this is it's just good practice with really understanding what a CDF is. Cuz the better you understand CDFs, then
the easier it is to see why this is true. So to prove this, all we need to
do is to compute the CDF of X. This notation means that X has the CDF F,
that is, X follows this distribution. So all we have to do is compute the CDF
of X, but that's actually pretty easy. Because by definition, X is F inverse
of u, I'm just plugging in what X is. Now let's apply capital F to both side. So I am just putting F here and F here. And that's an equivalent because I made
these nice assumptions about F that's an equivalent to n,
u is less than and equal to F of x. You know if we didn't have
an increasing function then if I apply both sides by minus one then
the inequality flips, things like that. But since we have an increasing
function then It's preserved. And since its invertible, this is really the same event
just written in a different way. Now we are done with that, because what
the probability that u is less than or equal to F of x. I'll just draw a simple little picture. U is uniform from 0 to 1, and F of x,
remember that's a probability, so that's just some number between 0 and
1, let's say it's there F of x. Now I said that probability is
proportional to length for a uniform and in this case that proportionality
constant is just 1 because the length of the whole interval is 1. So for uniforms 0 and 1, the probability
of an interval is its length. So we want to know, what's the probability
that u is between here and here. That's just the length of
the interval that's F of X. And that's the end. That's the end of the lecture. So have a good weekend. Thanks.

subset: commoncrawl


text:
The students and faculty will work individually and collaboratively, through a program of reading, problem solving, discussion sessions, workshops, lectures, and field and laboratory studies, to achieve the basic goals of this program:
To gain understanding of the scientific principles and concepts of introductory geology, botany, biology, quantitative methods, political economy, reading, writing, speaking, and other related subjects; and to develop problem solving skills, in the class, field, and laboratory as required
A. Come on time to all program meetings, with any assigned readings, problem sets or written work completed.
B. Participate fully in group activities, including workshops, labs, and field studies.
C. Inform faculty promptly of difficulties, confusions, problems, etc. with any aspect of the program.
D. Write a detailed self-evaluation that reflects achievement in the program at the end of the program.
E. Write an evaluation of each faculty member (separately) and deliver them to the faculty member or to the program secretary by the time of the evaluation conference at the end of the quarter.
A. Each student who regularly attends and participates in lectures, labs, field studies, and other scheduled activities, who completes the assigned written work, who completes and does adequately on the exams, and who turns in a self-evaluation, all with acceptable content and quality, can expect to receive full credit.
B. Any student whose work appears not to be meeting standards for credit will be so informed by his faculty seminar-leader in writing by the end of the fifth or sixth week of the quarter with suggestions for improvement. Students who do not receive a written mid-quarter warning may nevertheless be denied credit based on unsatisfactory performance after the fifth week.
C. Any student who plagiarizes material through failure to attribute and acknowledge sources or through failure to acknowledge joint authorship, or who cheats on quizzes or exams is liable to lose all credit for the quarter.
A. Prepare for and guide learning from discussions of the texts and seminar books, from the laboratory and field studies, from group and individual projects and from other assignments.
B. Oversee the program reader's review of students' written work so it is read, commented upon, and returned in a timely manner.
C. Respond to students' questions and concerns about the material, pace, and/or organization of the program.
D. Be available at least one hour a week ("office hours") for individual student concerns.
E. Keep students informed about their progress and notify each student by the end of the fifth or sixth week of the quarter if his/her work is not satisfactory.
F. Provide for a written evaluation of each student's work at the end of each quarter following the guidelines of section 7.620 in the Faculty Handbook.
return to IES Information
made by J. Stroh; Aug 8 1996


text:
CHAMBERLAIN — A Chamberlain man was charged Jan. 13 with choking a woman and fighting with her in front of a 6-year-old girl.
Matthew Sherman, 30, was convicted of assaulting the same woman in front of the same child in October and now faces a Class 3 felony aggravated assault charge and a Class 1 misdemeanor charge of contributing to the abuse, neglect or delinquency of a minor.
Chamberlain police spoke to the alleged victim at Dakota Counseling Institute. According to court documents, she said that at about 3 a.m., she and Sherman had gotten into an argument that began with her waking him up in the passenger seat of a car where he had fallen asleep.
The woman said Sherman pushed on her leg inside the vehicle and she hit him in the face when he would not stop. Sherman allegedly then climbed on top of the woman and began choking her. The woman broke the windshield with her foot and was able to get away from Sherman and into the house, where she dead-bolted the door and Sherman reportedly broke the door frame to get inside.
The woman reportedly laid on a couch in the fetal position and said Sherman would push her down when she tried to get up, punching her in the head at one point. The 6-year-old girl was awakened during the incident and came into the room where Sherman and the woman were.
The woman eventually left the house and stayed in her car until she was able to go to DCI, where another woman contacted police.
Sherman reportedly said the woman had hit him and that he had been trying to defend himself. His knuckles and hands were found to be bruised, scraped and swollen, which he allegedly said was from hitting a wall and the car. Bruises on the woman's arms, legs, hands, neck and chest were reportedly observed by police.
Sherman was charged with aggravated assault in August 2019 for allegedly choking the same woman. He and the victim were both given suspended sentences after both pleaded guilty to contributing to the abuse, neglect or delinquency of a minor, and Sherman was also given a suspended sentence for simple assault.


text:
Unlocking human potential
" Don't be afraid to give up the good to go for the great "
Investors in Excellence is a globally-recognised performance improvement, organisational development and sustainable growth specialist.
Our expert team helps forward-thinking organisations to deliver continuously-improving outcomes that are linked directly to business goals.
Working in partnership with you, we'll share the best practice that we've gained over the last 24 years through working with a host of large, medium and small organisations in the private and public sectors, both in the UK and internationally.
We know what works and we will share our knowledge with you.
News & Customer Stories
Supporting those most affected
We're connecting with you …In these challenging times, developing and learning can't be forgotten when working remotely away from school and workplace is something very new for most people. Adaptability, agility and flexibility is the core of working excellently. https://www.investorsinexcellence.com/wp-content/uploads/2020/03/20200324-CHALLENGING_TIMES.pdf [...]
Two IiE workshops at the Southern Manufacturing Show
Inspiring insights into excellence will be provided by IiE's CEO Judy Hart at the Southern Manufacturing and Electronics show taking place at Farnborough (Feb 11-13). Judy's seminars, designed to help organisations to optimise their performance, follow fantastic feedback for her sessions at last year's event. Seminar 1 – If it's [...]
Website Launch
Base 8 is thrilled to announce the launch of the new website for the globally recognised organisational development, continuous improvement and sustainable growth specialist firm 'Investors in Excellence' (IiE). We have been working in collaboration with IiE for the past 2 years, having been privileged to be selected as their [...]


text:
Tanning | Homesteading Skills Every Homesteader Must Be Well Equipped
10 Trees Every Homesteader Should Plant (And Why)
Planting certain trees around your homestead can offer benefits. These include trees that provide food, medicine, firewood, and more. #homesteadsurvivalsite #trees #medicinal #homesteader
4 Tools Every Chicken Keeper Needs
Keeping chickens in your backyard can be a burden as much as it can be fun, we know that, so here are 4 tools every chicken keeper needs. #chickens #backyardchickens #chickenkeepers #chickencoop #paintscrapper #flashlight #homesteading #farmlife
Why every flock needs a good rooster
The benefits of having roosters with your flock
Raising Quail Is For Every Homestead [Video Tutorial] | Pampered Chicken Mama: Raising Backyard Chickens
Raising quail for sustainable eggs and meat is easy. If you live in an urban area that outlaws chickens, quail are a good alternative. Here's how to start quail farming. From FrugalChicken
6 Reasons Every Homesteader Should Own a Dutch Oven
Dutch ovens are durable, versatile, easy to clean and maintain, and an all around great investment. Here are a few more reasons you should get one. #homesteadsurvivalsite #homesteading #dutchoven
10 Trees Every Homesteader Should Plant (And Why)
Homestead gardens, Trees to plant, Fruit garden, 10 tree, Fruit trees, Urban homesteading – 10 Trees Every Homesteader Should Plant (And Why) – #Homesteadgardens
6 Skills Every Homestead Lady Should Learn ~ Making It Home
What are the 6 top skills every woman of the homestead should learn? I go over what skills are extremely valuable to have for the women on the homestead striving for a more self-sustainable life for their family.
145 Homesteading Skills Every Homesteader Must Be Well Equipped
Honey | 133 Homesteading Skills for Beginners
27 Real Part-Time Night Jobs from Home (Make $1000 Every Week)
looking to work late at night from home? there are many late-night "work from home" jobs you can choose from which are both flexible and lucrative. Legit work from home jobs, online jobs, part time work from home jobs, money making side hustle ideas, late night jobs to earn extra cash on the side. If …


text:
Robert Alexander "Bob" Spiers (27 September 1945 – 8 December 2008) was a director. He is particularly noted as the director of the early series of Absolutely Fabulous (1992–2001), the musical comedy Spiceworld, and of the second series of Fawlty Towers (1979). He also worked with Steven Moffat on Press Gang and Joking Apart. Spiers won a BAFTA Award twice for his work as a television comedy director.
Born in Glasgow, Scotland, he attended Southgate College in the 1960s. "Jock", as he was affectionately known at the time, organised several student trips from the college to mainland Europe, including Brussels and Cologne, during this period. Spiers joined the staff of the BBC in 1970, working as an assistant floor manager and later a production assistant, before eventually working his way up to become a director and producer. In this capacity he worked on several high-profile programmes, such as Dad's Army, Are You Being Served? and It Ain't Half Hot Mum, all sitcoms, a genre with which he became particularly associated. During the 1970s he met his wife, Annie.
It was because of his talent for directing comedy that he was chosen to handle the second series of Fawlty Towers in 1979, which already had an enormous reputation on the basis of its initial six episodes in 1975, and it won him his first BAFTA award. Spiers subsequently provided a commentary for all the episodes he directed of Fawlty Towers when the series was released on DVD. Shortly after he directed the series and the unbroadcast pilot of Not the Nine O'Clock News, Spiers left the staff of the BBC to work as a freelance director.
Throughout the 1980s he worked on a number of programmes, of particular note being Channel 4's anthology comedy series The Comic Strip Presents... and the BBC sketch shows French and Saunders and A Bit of Fry and Laurie. He began his association with writer Steven Moffat in 1989, directing over half of the episodes of the teen comedy drama series Press Gang (1989–1993) for the ITV network. According to Moffat, Spiers was the "principal director" taking an interest in the other episodes and setting the visual style of the show. Spiers particularly used tracking shots, sometimes requiring more dialogue to be written to accommodate the length of the shot. The other directors would come in and "do a Spiers". Spiers then directed all twelve episodes of Moffat's sitcom Joking Apart (1993, 1995). The show won the Bronze Rose of Montreux and was entered for the Emmys.
His association with comedians Dawn French and Jennifer Saunders from directing their sketch show led to him working on individual projects from each of them during the 1990s. With French he worked on the macabre comedy anthology series Murder Most Horrid, with some episodes written by Moffat (such as "Overkill"). More famously, with Saunders, he helmed Absolutely Fabulous throughout the decade, the show having originated in a sketch from an episode of French and Saunders which had also been directed by Spiers. Absolutely Fabulous won Spiers his second BAFTA award.
In 1997 he directed the pop group the Spice Girls in their film Spiceworld. He had been working in America on the Disney film That Darn Cat at the peak of the Spice Girls' popularity, and was unaware of the group when first offered the job of directing Spiceworld until friend Jennifer Saunders advised that he take it. He arrived at a meeting with them in a New York hotel unaware of what they looked like. He also directed two episodes of the Australian ABC series The Adventures of Lano and Woodley.
He died in December 2008 in Widecombe, Devon at a family home after a long illness. He owned a restaurant in Ashburton, Devon called Number 14, which still displays both of his BAFTA awards.


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Raspberrry Pineapple Smoothie
It's time for another recipe remix – where we take complicated creations and simplify them. Helping us out as usual is Theresa Toth from Easy, Peasy Eats. Today we're learning about simple smoothies – both delicious and healthy.
Courtesy – Theresa Toth
- 1 cup pineapple, chopped
- 1 cup raspberries
- 1 cup orange juice
- 1/4 cup cilantro
- 1/2 cup ice
- Place all ingredients into a blender and pulse until smooth.


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Category Archives: News Item
With the studio temporarily closed we have some fun things planned for everyone while we are isolating. In order to participate you will need to have access to Zoom, a video conferencing software. Just download the app for the platform that you have: For Zoom on your PC computer download the first option at: https://zoom.us/download#client_4meeting…Read more
We have uploaded a short video to practice your Cha Cha technique with Barbara Lynn on the Arthur Murray Coquitlam VLog Page...Read more
We are continuing our closure for this week due to the Covid-19 outbreak in our area. There are some online resources posted on the Arthur Murray Coquitlam Facebook page, on our Instagram channel and on our Vlogs page on the website... Arthur Murray Coquitlam Vlog PageRead more
We are continuing to monitor the press conferences from the BC Health Ministry and there are further restrictions on our interactions mandated in today's March 17th update. They are asking all businesses to maintain a social distancing of 1 to 2 metres for anyone coming in. As of March 20, 2020 Arthur Murray Coquitlam…Read more


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In Sinews of War and Trade (Verso) Laleh Khalili, professor of politics and international studies at SOAS, University of London, explores the vital importance of ports and shipping in defining the new battlegrounds of the 21st century. In particular, she argues, the Arabian Peninsula is the crucial nexus for the transportation of the goods and commodities that define and enable contemporary capitalism, inevitably making it a region of compelling strategic importance and of military and diplomatic conflict.
She will be in conversation about her book with Dr Rafeef Ziadah, who is conducting research on the commerce and politics of the Gulf region at SOAS.


text:
Truelancer is a curated freelance marketplace with thousands of top rated Banner Design Freelancers in Sri Lanka. It is simple and quick to Post your job and get quick quotes for your Sri Lanka Banner Design Freelancers requirement.
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Enjoy a scenic 30 minute sightseeing tour over Hoedspruit in a microlight - you might even catch sight of the Big 5!
- Hoedspruit (Hoedspruit)
- Gift Voucher valid for 1 Participant
Enjoy a scenic 30 minute sightseeing tour over Hoedspruit in a microlight - you might even catch sight of the Big 5!
Looking for a specific date rather than a voucher? Wherever you see the Book Now banner, we can check live availability and make reservations for you!
Take to the skies with this thirty-minute microlight flight over Hoedspruit in Limpopo province. It's a flying experience you'll never forget!
Flights take off from the Hoedspruit Civil Airfield, where you'll be met by an experienced pilot. They'll brief you on what to expect before leading you to the lightweight, fixed-wing aircraft. After taking your seats, the pilot will prepare for take-off—and then it's up, up and away! If you've never flown in a microlight before, a thirty-minute flight is the perfect introduction. Once you're at altitude, the pilot will offer a brief flying lesson, allowing you to take the controls yourself. Alternatively, you can choose to sit back and simply enjoy the views. A variety of game can be spotted on the ground below, including members of the Big Five. Lions and leopards can be elusive, but you've a good chance of seeing elephants, rhinos and buffalos—so be sure to keep a lookout!
The 30-Minute Microlight Flight in Hoedspruit would make a thoughtful gift for visitors and locals alike, and anyone wanting a unique game viewing experience. Make someone smile with an Experience Days gift certificate today!
30 minute microlight flights can be booked on weekdays and weekends throughout the year, subject to weather and availability.
Minimum age: 5
Maximum weight: 110kg
Experience Duration: 00:30
Time Spent at Venue: 01:00
The duration of the flight is 30 minutes.
Participants: The Half Hour Microlight Flight in Hoedspruit is for 1 participant.
Max Group Size: 1
Spectators Allowed on Site: Yes
Participants will fly one at a time with the pilot in the front seat.
Spectators are welcome to watch from the hangar.
Please wear comfortable clothing, and bring a jacket as it can get chilly in the air.
In bad weather the flight may be scheduled for another date.


text:
Cellucity Pty Ltd
Cape Town
South Africa - 8005
Customer Service
|+27 214 011 300
|1 0
|+27 214 011 300
|1 0
I was today (14/02/19) in canal walk at cellucity for a powerbank and told the sales person that I need one that can super charge my huawei p20 pro in 1 hour. I wast told that the...
Hi, we were in south africa/zimbawe between 12-24 december, and had rented a wifi router from the airport for usage.
Over that 12 days, we only used instalgram and whassap apps. However, there was an incredible charges of 18gb made to our usage, including 10gb when we were spending time in zimbawe and did not even use the router (as we were informed by the airport staff that the router will not work).
We had feedback this to cellucity (nurjehan mohamed) who informed us that they will investigate further. However, we have not heard from her since 2 january 2019, and have since been charged s$300 to our credit card bill.
I have highlighted to her that this usage is impossible and to suspend the billing until investigation is completed. However, I don't believe she has put any attention on this case so I am raising this as an official complaint on this case and against nurjehan.
Name: lim zhi han derrick / goi ching rong
Id: s8531263d / s8502158c
Stay in south africa: 12 december 2018 - 24 december 2018
Card: [protected]
Dear Mr Rong
Thank you for the post,
I can see that Nurjehan has completed the request for data information forms and has sent this to the Network on 28 December 2018.
Please do understand that a request for further information does take a few weeks to obtain however also that the departments within the Network that these requests are made to are closed at the end of December and only reopen the 2nd week of January,
I can also confirm that no data was billed for for your roaming period in Zimbabwe.
Nurjehan will be in touch as soon as the full report is received back as to what apps the data was used on however billing still remains until the Network states that the data was incorrect which in most cases is not the case.
Devices often use data to back up between apps as they generally all run in full in the background and its often best to turn all apps off completely to avoid high data usage.
Nurjehan will be in touch.
Customer Service has done everything in their power to resolve the complaint. All attempts to contact the customer have failed. Therefore, the complaint has been annulled and must not be considered where image of the company in question and its services are concerned.
Good day I make reference to the south african national consumer protection act, 68 of 2009, which came into effect on 1 april 2011. The consumer protection act (cpa) aims to...
Hi I tried to take out a new line and was credit check over 15 times in 3 days. Stay far away from this company. Cellucity bemoreshopping center (sandton) Beside dropping my...
Incompetent and dishonest staff and manager Upgraded my note5 to a note8 in october at cellucity pavillion branch... By december my screen cracked. I then went to samsung to sort...
I have now complained on every level to vodacom and cellucity where I purchased my contract cell phone.
To cut a long story short, I recently upgraded my contract with vodacom and even paid in a huge sum of monies in to ensure my monthly premium is as low as possible, and to my surprise my premium was whooping r946!! Instead of the 400 rand, I made contact with vodacom and the franchise "cellucity" I also decided to call into the call centre to ask what's going on. I then explain to the call centre agent that I get 250 megs free with my package and I bought an additional bundle package of 250m in order not to incur additional charges. The call centre agent tell me that the 250megs what was advise to me once the phone was activated only take place at the end of the month and I now have to pay for the charges incur. The branch manager then calls me from cellucity take advise me that the sales lade was incorrect and I need to provide my bank details to credit my account and that cellucity will be liable... Days lays he calls again stating that they not paying because the sales person said the sales lady is now turning her story and I have to pay up. This is yet another poor reflection of your inadequate service delivery and it appears that no one at "vodacom and cellucity" cares a hoot about my problem. I did what I needed to safe guard myself and now they pushing me from pillar to post. I've been extremely patient with reference to incorrect billing on my account and they said they will correct it and not changing their story. Take note because there could be countless other people out there and it may happen to you
I let cellucity in cavendish square sell me a nokia e75 phone for r4500 in early march. The sales people were very good at telling me about the guarantees, and the fact that my phone would be replaced if there was any problem in the first seven days. What they didn't mention was that if you take your phone back for replacement, they will actually only replace the phone when some faceless technician at 'vodacare' gives them the go-ahead. So they send the phone away to 'vodacare' and then renounce all responsibility for how long it may take to get your phone back.
Apparently this was in the fine print I signed. Well, they might win a court case against me, and bully for them if they do. There's a difference between covering yourself legally and actually using that fine print as an excuse to keep a slack, rubbish policy whereby a customer can bring a malfunctioning phone back and be left for a limitless amount of time to get a replacement. For me it's nearly a month and counting now, and I suppose it could take years. My phone may be obsolete by the time it comes back in 2014... What service!
A month ago I purchased a dell studio from celluctiy in cavendish square. On the day of purchase I had to return it three times as it had aconnecion poblem, tht even their 3g technicians struggled with. The pc has been returned about six times since purchase date, and as i'm typing this, I have check every single letter to ensure it is there. It misses at random.
I have mailed the ceo mr sean joffe, explaining my frustrations, and he dismissed it as 'it's simle to fix the homepag.. Just click on tools etc ect (ignoring the fact that I allready mentioned that the problem returns after two days)', or 'we could send a dell technician to you'. I have been in contact with dell about the problems, who forwarded links for downloads to me. I manage to do the downloads slowly and with much effort, as the connection kept dropping.
I demanded a relacement pc for this 'obiously' dudd machine, but t fell on deaf ears.
Potential buyers should be aware of this poor quality product, and dell cellucity's refusal to accept responsibility for their product.
Please note :spelling mistakes are missing letters caused by this pc!!
On the 31st july, 2006, I purchased a new contract with a company called 'cellucity' at century city shopping mall, cape town, for a cellular phone a nokia 6280. The cell phone was connected on the 2nd august and on the 3rd august I noticed a big leaf shaped shadow on the screen. We checked to see if we had made some mistake but realized that there was a fault with the phone. When we took the phone back, we were told that we must have broken the phone. I know that the phone was never abused as I never even had a chance to even put one telephone number on the phone as I was still using my old phone which I have had for 3 years and have never had a days problem with it. Anyhow cellucity took the phone in and on the 8th august and sent it in to 'vodacare' to have it checked. They tell me that it will cost us$70.00 to repair. I told 'cellucity' that I have not done anything to the phone. They became rude and aggressive and did not want to help me any further. To this day, I still have no cell phone and feel that this sort of service in such a highly competitive world is disgusting and a very bad reflection on nokia who should check out their distributors as they are causing them to loose business. Incidentally, I have also been threatened that I will be blacklisted if I don't pay the contract which is for 2 years.
John, cape town, south africa
Cellucity at century city shopping mall, cape town


text:
Dumpster Rental in Saint Francis AR
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Dumpster Rental Companies Near Saint Francis
|Company Name
|Cunningham Metals Inc
|(479) 968-1574
|JCMC Trash Services
|(501) 440-1221
|Dewpoint of Arkansas
|(501) 847-7555
|Bio Waste Disposal
|(479) 202-5277
|Suburban Sanitation Inc
|(501) 767-9451
|Brandon's Trash Services
|(479) 857-5590
|Courtesy Sanitation
|(479) 267-6224
|Iesi AR Midway Division
|(870) 481-8600
|D C Trash Services
|(479) 890-6269
|Frac N Vac Tanks
|(870) 862-1602
Our Service Area
Zip Codes in Saint Francis72464
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Dumpster Sizes Available Saint Francis
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10 Yard
Great for small projects
- 12 feet long
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For moderate sized cleanouts
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text:
Amazing opportunity to own a charming 2-bedroom 1-bathroom home in one of Chico's favorite neighborhoods with vintage homes and mature trees. Close to Downtown, Campus, Enloe, and neighborhood schools. This updated 1930s classic with oak hardwood floors still has a milk bottle delivery door in the wall in the dining room! There is a nice covered front porch to watch the world go by, and a private backyard with a beautiful giant shade tree. A long concrete driveway beside the home leads to a detached garage ( 16' by 18' ). This long driveway with double gates could nicely accommodate a modest sized RV. The fireplace in the living room has a wood stove insert that keeps the home warm and cozy. Off the kitchen are a large laundry room and a large walk-in pantry! Don't miss this one!
|1,188 SqFt
|Lot Size
|0.14 Acres
|Year Built
|Property Type
|Residential Income
|Listing Number
|Listing Agent
|Olivia Larrabee
|Listing Office
|Century 21 Select Real Estate, Inc.
|Total Building
|Total Units
Listed by Olivia Larrabee of Century 21 Select Real Estate, Inc.. Listing data last updated 4/5/2020 at 2:37 AM PDT.
Based on information from California Regional Multiple Listing Service, Inc. as of 3/15/2020 4:30 PM. This information is for your personal, non-commercial use and may not be used for any purpose other than to identify prospective properties you may be interested in purchasing. Display of MLS data is usually deemed reliable but is NOT guaranteed accurate by the MLS. Buyers are responsible for verifying the accuracy of all information and should investigate the data themselves or retain appropriate professionals. Information from sources other than the Listing Agent may have been included in the MLS data. Unless otherwise specified in writing, Broker/Agent has not and will not verify any information obtained from other sources. The Broker/Agent providing the information contained herein may or may not have been the Listing and/or Selling Agent.
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constraction steel h beams
Mar 26, 2018· Types of Construction Beams & Their Uses ... Wooden beams are traditional, but steel versions are becoming more common as homeowners turn unfinished or outdated attics and lofts into modern living spaces. Cantilever Beams. You'll find cantilever beams in projecting house elements such as bay windows, balconies, roofs and canopies. ...
Column and Beam system in construction - Basic Civil ...Introduction
Types of Construction Beams & Their Uses Hunker
Mar 26, 2018 · Types of Construction Beams & Their Uses ... Wooden beams are traditional, but steel versions are becoming more common as homeowners turn unfinished or outdated attics and lofts into modern living spaces. Cantilever Beams. You'll find cantilever beams in projecting house elements such as bay windows, balconies, roofs and canopies. ...Author Elizabeth Michaelson Monaghan
Sizes for Steel I-Beams, HSS, Channel, and Angle [+Free ...
Aug 21, 2017 · Sizes for Steel I-Beams, HSS, Channel, and Angle [+Free Calculator] ... Steel I-Beam Sizes. I-Beams are also known as H-Beams, W-Beams (for "wide flange"), Universal Beams (UB), Rolled Steel Joists (RSJ) or Double-T. ... Steel I-Beams are typically produced by a rolling process that was invented in 1849 by Alphonse Halbou in France. The use ...User rating 5/5
Beam (structure) - Wikipedia
Most beams in reinforced concrete buildings have rectangular cross sections, but a more efficient cross section for a beam is an I or H section which is typically seen in steel construction. Because of the parallel axis theorem and the fact that most of the material is away from the neutral axis , the second moment of area of the beam increases ...
What is Beam Construction? (with pictures) - wisegeek
Sep 13, 2019 · Steel-reinforced beams generally have a steel core through the middle and/or a series of metal bands wrapped around the outside. These are common in extremely heavy or high-traffic buildings. Steel beam construction revolves around two types of beams—the vertical round beam and the I-beam.
Construction Steel H Beam -
Construction Steel H Beam , Find Complete Details about Construction Steel H Beam,Construction Steel H Beam,Galvanized Steel Beams,Hollow Steel Beam from Steel H-Beams Supplier or Manufacturer-Bochi CorporationUser rating 5/5
column and beam construction tips - Ask the Builder
Column and beam construction is common in most homes. Beams can be as short as 3 feet or as long as 16 feet. Don't guess sizes. Consult with a professional engineer. Every week I get an email from a homeowner who wants to tear down a wall or she/he's contemplating some other …Column and Beam Construction Tips1. Don't guess - get an engineer 2. Concentrated loads important 3. Column and beam construction VIDEOS BELOW 4. Steel is king 5. CLICK HERE to Get...Column and Beam Construction VideoHere's a great video showing decorative wood columns and beams. Simple ones are made with 2x4s and 2x12s! Related LinksRemoving a Load Bearing Wall...Continuous Load Path For Beams & ColumnsStructural engineers and great builders know all about continuous load pathways. This is a fancy phrase for creating a solid bearing pathway from t...Steel Is King For Columns & BeamsThe strongest of all beam materials is steel. You can span greater distances given the same beam height using steel in lieu of any other material.T...
What are beams, slab, footings & columns in construction ...
Dec 22, 2016 · BEAMS:- a long, sturdy piece of squared timber or metal used to support the roof or floor of a building. or A beam is a structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the mate...What is the standard size of a column for building?Sep 13, 2019What are the different types of columns in construction?Apr 25, 2018What is the difference between a beam and a column?Mar 13, 2018What is a clear cover for a slab beam column footing?Oct 10, 2017See more results
A completely new version of the Steel construction - Cost supplement has been produced to provide construction professionals with the latest information on costing structural steelwork...(08/05/18); Updated guidance on the Cost of structural steelwork has been published by AECOM. Costing Steelwork #5 article published in Building Magazine presents the latest cost data for steel frames, and a ...
Steel Construction Manual American Institute of Steel ...
The v15.1 Companion to the AISC Steel Construction Manual is a resource that supplements the 15th Edition Steel Construction Manual and is keyed to the 2016 Specification for Structural Steel Buildings. The v15.1 Companion is an update of the v15.0 Design Examples with the design examples and tables split into two separate volumes.
Steel Construction Manual American Institute of Steel ...
The v15.1 Companion to the AISC Steel Construction Manual is a resource that supplements the 15th Edition Steel Construction Manual and is keyed to the 2016 Specification for Structural Steel Buildings. The v15.1 Companion is an update of the v15.0 Design Examples with the design examples and tables split into two separate volumes.
Structural Steel in Construction from Construction ...
Hot rolled steel shapes, most commonly found as steel beams and columns on construction projects, are created in steel mills by rolling the heat steel between large rollers, deforming the steel into the typical shapes W, S, C, angles, tube sections, pipes, etc. Most hot formed steel is either 36,000 psi or 50,000 psi yield strength.
usually support beams and columns. Grade – a designation of the ASTM that identifies the chemical composition and strength characteristics of structural steel, e.g. ASTM A50 which identifies the steel as having a yield (or failure) strength of 50,000 pounds per square inch.
How to Estimate the Cost of Different Structural Beam …
Steel components are usually measured in SF and LF but can ... I-shaped, H-shaped and channels designate the weight in the sizing such as W12x16, S8x23, HP8x37 and C10x30, 5 P a g e where 16, 23, 37 and 30' represent the weights in lbs. For channels, angles and for hollow steel sections (HSS), the
Journal of Constructional Steel Research - Elsevier
The Journal of Constructional Steel Research provides an international forum for the presentation and discussion of the latest developments in structural steel research and their applications. It is aimed not only at researchers but also at those likely to be most affected by …
Typical Steel Connections - Memorial University of ...
Steel Connections -Dr. Seshu Adluri Introduction Steel Connections Many configurations are used for force transfer in connections. The configuration depends upon the type of connecting elements, nature and magnitude of the forces (and moments), available equipment, fabrication and erection considerations, cost, etc.
Steel construction products - SteelConstructionfo
Asymmetric steel beams are often used as part of a shallow floor system. Their key feature is a wider bottom than top flange, which enables the slab to sit on the upper surface of the bottom flange with adequate bearing, ... ↑ 2.0 2.1 BS EN 10365:2017 Hot rolled steel channels, I and H sections. Dimensions and masses.
Steel H-Beams - weiku
Steel H-Beams Products catalog and Steel H-Beams manufacturer directory. weiku is a Steel H-Beams trade platform offering information about Global Steel H-Beams Products and global Steel H-Beams …
Wood Construction Connectors Simpson Strong-Tie
Strength Beyond Steel. From testing to service to availability, everything we do is engineered to ensure your project's success. With every product, you know you are getting the quality, value, service and on-time delivery that we have built our reputation on for the past 60 years. ... Our full line Wood Construction Connectors Catalog includes ...
Bar Bending Schedule [BBS] Estimate of Steel in Building ...
Jan 12, 2019 · Bar Bending Schedule [BBS]:-Before dealing with the BBS, it's very important to learn the basics of Bar bending schedule. The below-mentioned table is a kick-start guide for learning Bar bending schedule from scratch. (If you are viewing the below table through mobile, scroll horizontally for a …
Type of Steel Bars - Mild steel bars, deformed steel bars ...
Mild steel bars are used for tensile stress of RCC (Reinforced cement concrete) slab beams etc. in reinforced cement concrete work. These steel bars are plain in surface and are round sections of diameter from 6 to 50 mm. These rods are manufactured in long lengths and can be cut quickly and be bent easily without damage.
How to Use Steel Metal Deck Support Posts and Columns ...
The most cost effective steel post is 4.5" diameter with 1/8" thick walls. Many of these are part of an assembly for a decorative columns. Steel is rarely used in residential deck construction because it is expensive, not readily available and more complicated to install.
Reinforcing Steel Placement Handbook
The steel in the compression zone does not significa ntly increase the compressive strength of the concrete and therefore the area above the neutral axis is assumed to be solid concrete for design purposes. Placing the reinforcing steel with more than the design cover causes the neutral axis to be shifted h igher in the section, which
Steel frame - Wikipedia
Steel frame is a building technique with a "skeleton frame" of vertical steel columns and horizontal I-beams, constructed in a rectangular grid to support the floors, roof and walls of a building which are all attached to the frame.The development of this technique made the construction of …
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