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Let $Y=(Y_1,Y_2,\ldots,Y_n)^{T}$ and $Y_1,\ldots,Y_n$ are independent normal random variables with mean $0$ and variance $ \sigma^2$. Find the distribution of the following statistics and give your reasons

a) $T=\displaystyle \sum_{i=1}^{6}(Y_i-\overline{Y})^2$

b) $U =\displaystyle\frac{\sqrt{5}\times Y_6}{\sqrt{Y_{1}^2+Y_{2}^2+Y_{3}^2+Y_{4}^2+Y_{5}^2}}$

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T/σ is chi-square 5 df (sum of squares of 6 dependent standar normal variables) U is student t 5 df (normal divided by independent estimate of standard deviation)

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    $\begingroup$ Michael: I usually vote up your answers, because they are neat and useful, but I have to admit I do not like it when you answer homework questions. This is because you just give the right answer and don't let the OP think about the answers. If you don't mind, I would suggest you to give hints better than the answers: in that way, you give the OP the opportunity to THINK. $\endgroup$ – Néstor May 18 '12 at 8:24
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    $\begingroup$ totally agree with Nestor, answers should be replaced by hints so students like me have a chance to familiarize with the materials. Anyways, anything would do then. $\endgroup$ – Viet Hoang Quoc May 18 '12 at 9:05
  • $\begingroup$ I would not generalize to say that I just give answers to homework problems. In this particular case I did. I am not sure what a good way to hint on this is and I think how anything is answered on any question is a judgement call. I gave the solutions but did not derive it or cite any theorems. The OP may be required to do that and if so he needs to do a derivation and this at least lets him know what the solution should be in the end. My answer was very brief and leaves a lot of work that the OP could do. $\endgroup$ – Michael R. Chernick May 18 '12 at 11:17
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    $\begingroup$ @Viet Hoang Quoc Do you really think that because I told you what the distributions are that it leaves nothing for you to do for this problem? I think you still should derive the solutions or cite theorems that do it. The chi square and t distributions are important for inference and the way the question is formulated you might not notice that these statistics are in the form for cho sqiare and t. maybe I gave a big hint but I think it is still far from a complete solution to the problem. $\endgroup$ – Michael R. Chernick May 18 '12 at 11:32
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    $\begingroup$ A constructive way to provide hints, Michael, is to respond with questions of your own. (The biggest risk in posing counter-questions is that it may cause some people to think you don't understand the subject, but what does that matter? You don't have to prove anything here.) I like your previous comment because it points out that just obtaining an answer is usually of less value than knowing why it is correct. Including some version of that comment in your reply here would improve it and, I suspect, might satisfy the concerns of the other commenters. $\endgroup$ – whuber May 18 '12 at 14:57

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