Please ask questions about the first assignment (due October 14) here.

If you know the answer to a question, or part of the answer, or where to find the answer, please do post a reply!

For the first problem, I was wondering what is necesarry for proving that the distribution if the means is normal.

It is simple enough showing that the mean and variance is as stated, but I have a nagging suspicion there is more to the problem...

Eric, you're right. You need more than the mean and the variance of the $\overline{x}$. I know of no easy way to get there other than using moment generating functions.

Edited by author 10-10-2002 09:27 PM
I don't immediately know how to prove that the sum of two Gaussian random variables is Gaussian. However this result is well-known :-) so there must be proofs of it published.

For this problem (and all others) you may use any and all published references. You should cite any references you use, and if you use an existing proof, you should write it up in your own words, as part of mastering it.

The proof is rather simple. If you compute the moment generating function for both the normal distributions and multiply them, you'll see that the product is the mgf for a gaussian. Since the mgf(x1+x2) = mgf(x1)*mgf(x2) a little number crunching will prove the result.

I was wondering about how detailed our proofs have to be. Where should we draw the line as to what to simply cite and what to actually prove?

If it is a well-known result that the sum of two normal distributions is a normal distribution, and (in my opinion) lesser known results about moment generating functions, then why is it a better proof to use the moment generating function theorems than just to assume that the sum of two normally distributed r.v.'s is normally distributed?

Edited by author 10-11-2002 12:48 PM
I'm not sure that the MGF proof is not as simple as you wrote. You show that MGF(Gaussian) ---> some form. Then you show that
MGF(Gaussian+Gaussian) ---> same basic form. But how do you know that there is no other distribution F such that MGF(F) ---> form for a Gaussian?

Another way to prove the homework (I think) is if you first prove a couple of things about normals:
  Let: X ~ N(\mu_1, \sigma_1^2)
       Y ~ N(\mu_2, \sigma_2^2)
  Then: X+Y ~ N(\mu_1+\mu_2, \sigma_1^2+\sigma_2^2)
  Also: cX ~ N(c\mu_1, c^2\sigma_1^2)
I think the first can be proved through the "convolution theorem" which is relatively easy to prove with a bit of calculus, and the second claim can also be proved fairly quickly. (The convolution thm can be found in Grimmet & Stirzaker ("Probability and Random Processes").)

Here is a reference for a proof that the sum of two Gaussians is Gaussian that does not use MGFs: http://www-stat.stanford.edu/~susan/courses/s116/node114.html

Thera are quite a few mathematical results that are well-known and widely used, but whose proofs are not well-known, and/or difficult. It's ok to assume a well-known result when proving something else, but it's still good to know how it can be proved.

I don't want everyone to get stuck on this first problem. Please move on to the others and work on the MVUE problems!

According to "Introduction to Mathematical Statistics and Its Applications" by Richard J. Larsen and Morris L. Marx, on page 241, theorem 3.16.2:

Suppose W_1 and W2 are random variables for which mgf(W_1)=mgf(W_2) for some interval of t's containing 0. Then the probability distributions if the two random variables are identical.

Since these conditions hold for 1.1 a, it's safe to apply mgfs.

I have heard from several people who are working in study groups that the groups are making good progress on the first assignment. I really urge everyone not in a study group to form one!

How important is it to include the numerical examples from Matlab?