I have samples from normal distributions with the same mean but different standard deviations. I know the corresponding standard deviation for each sample. Is there a way to more accurately estimate mean other than taking an average over all the samples?
I have a large set of data points of the form (mean, stdev). I wish to reduce this to a single (better) mean, and a (hopefully) smaller standard deviation.  Clearly I could simply compute $\frac{\sum data_{mean}}{N}$, however this does not take in to account the fact that some of the data points are significantly more accurate than others.  To put it simply, I wish to preform a weighted average of these data points, but do not know what the weighting function should be in terms of the standard deviation.
How do I get an animated gif to work in WPF? Mathematical notation around the world What are the differences in mathematical notation around the world? I know that in some other countries they write 1,2 meaning 1.2, but what else can be confusing in an academic environment (when people are doing math on a board or on paper). Creating a DateTime in a specific Time Zone in c# I'm trying to create a unit test to test the case for when the timezone changes on a machine because it has been incorrectly set and then corrected.  In the test I need to be able to create DateTime objects in a none local time zone to ensure that people running the test can do so successfully irrespective of where they are located.  From what I can see from the DateTime constructor I can set the TimeZone to be either the local timezone, the UTC timezone or not specified.  How do I create a DateTime with a specific timezone like PST? What causes a force field to be "non-conservative?" Why do we say "is it" when asking a question rather than "it is"? What it says on the tin, a foreign friend of mine has asked and I can't tell him; apart from it sounding horrible.  For example:  "Why is it raining today?"   Instead of:  "Why it is raining today?" Why was the universe in an extraordinarily low-entropy state right after the big bang? Let me start by saying that I have no scientific background whatsoever. I am very interested in science though and I'm currently enjoying Brian Greene's The Fabric of the Cosmos. I'm at chapter 7 and so far I understand most of general ideas he has talked about. Or at least, I think I understand them :-)  There is however one part, at the end of chapter 6 that I can't grasp.  It is about entropy and the state of the universe a few minutes after the big bang.  On page 171 he says:     Our most refined theories of the origin of the universe -our most refined cosmological theories- tell us that by the time the universe was a couple of minutes old, it was filled with a nearly uniform hot gas composed of roughly 75 percent hydrogen, 23 percent helium, and small amounts of deuterium and lithium. The essential point is that this gas filling the universe had extraordinarily low entropy.   And on page 173-174:     We have now come to the place where the buck finally stops. The ultimate source of order, of low entropy, must be the big bang itself. In its earliest moments, rather than being filled with gargantuan containers of entropy such as black holes, as we would expect from probabilistic considerations, for some reason the nascent universe was filled with a hot,   uniform, gaseous mixture of hydrogen and helium. Although this configuration has high entropy when densities are so low that we can ignore gravity, the situation is otherwise when gravity can't be ignored; then, such a uniform gas has extremely low entropy. In comparison with black holes, the diffuse, nearly uniform gas was in an extraordinarily low-entropy state.   In the first part of the chapter Brian Greene explains the concept of entropy with tossing the 693 pages of War and Peace in the air.  At first, the pages are ordered. The specific order they are in make sense and are required to recognize the pages as a readable book called War and Peace. This is low entropy. It is very highly ordered and there is no chaos.  Now, when you throw the pages in the air, let them fall and then pick them up one by one and put them on top of each other, the chances you get the exact same order as the initial state are extremely small. The chance you get another order (no matter what order, just not the one from the beginning) is extremely big. When the pages are in the wrong order, there is high entropy and a high amount of chaos. The pages are not ordered and when they are not ordered you would not notice the difference between one unordered state and another one.  However, should you swap two pages in the ordered, low entropy version, you would notice the difference.  So I understand low entropy as a highly ordered state with low chaos in which a reordering of the elements would be noticeable.  I hope I'm still correct here :-)  Now, what I don't understand is how a uniform mixture of hydrogen and helium can by highly ordered? I'd say you wouldn't notice it if some particles traded places. I'd say that a uniform mixture is actually in a state of high entropy because you wouldn't notice it if you swapped some hydrogen atoms.  Brian Greene explains this would indeed be the case when gravity plays no important role, but that things change when gravity does play a role; and in the universe right after the big bang, gravity plays a big role.  Is that because a reordering of the particles would change the effects of gravity? Or is there something else that I'm missing here? product of normal and lognormal variates If x and y are uncorrelated normal variates, x*exp(y) will have a symmetric unimodal distribution with positive excess kurtosis. Has this distribution been named and studied? is there any diffrence to use nonlinear or liner activation function in single hidden layer network Why does the loss/accuracy fluctuate during the training? (Keras, LSTM) Is "education" a count noun? What are best practices for blocking UI when application is busy Decorate Tables You have $r$ red, $g$ green and $b$ blue balloons. To decorate a single table for the banquet you need exactly three balloons. Three balloons attached to some table shouldn't have the same color. What maximum number $t$ of tables can be decorated if we know number of balloons of each color?  I know the answer will at least be $\min(r,g,b)$. Also we need to add tables with $2$ reds, $2$ blues or $2$ greens. I cannot think of the formula for it Why can I see the output of background processes? For example, when I type ping 8.8.8.8 &amp;, why can we see ping process working? And when I type find / -name '*test*' &amp; , it's also seen on monitor.  Why is that? Is it not really in background? Making layer or group private/hidden/invisible in Layers Panel of QGIS I have a number of layers in the Layers Panel inside a group but I don't want them shown (there's quite a lot so it looks untidy). These layers are also joined to several others. Is it possible to make these layers invisible to the user (i.e. not shown in the Layers Panel, it is not about setting the visibility of the layer) but can still be joined to other layers, or perhaps to make the group invisible? Closest I found was this post but hoping there's a more direct approach: Total combinations from number of unique combination? How can you find count of the total combinations possible if you have the count of unique combinations? Suppose I need to form a sum of $3$ using $\{1,2\}$ Unique combn. $(1+1+1)$, $(1+2)$ hence '$2$' unique but total is '$3$' i.e, $(1+1+1)$, $(1+2)$, $(2+1)$ I know the elements used to make the combination Note: Number of unique combinations (ie. 2 here) is known. How can I find the total number of combination given the unique combinations? Using both “one’s” and “their” to refer to the same entity Proving that a function has a removable singularity at infinity Trying to install CPAN Compute integral $\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$ Compute the integral.   $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$  The answer at the back of the book is  $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$ I have 41 GB of space on my E:/ drive. Is it sufficient to install Ubuntu? My Windows XP is installed in C:/ drive. Is it OK for dual boot? Should I install Ubuntu (I am installing using Live CD). I don't want to lose my Windows option.