# Comparing the mean to the standard deviation

Let $X_1,X_2, \ldots ,X_n$ be i.i.d. random variables with normal distribution ${\cal N}(\mu,\sigma)$. Let

$$M=\frac{X_1+X_2+ \ldots +X_n}{n}, \ D=\sqrt{\sum_{k=1}^n (X_k-M)^2},\ Y=\frac{X_1+X_2+ \ldots +X_n}{D}$$

I have two related questions about those variables :

(1) Is it true that $M$ and $D$ are independent ?

(2) Is the distribution of $Y$ known ?

Those questions may be stupid as I don’t know much about probability.

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You should read about student's t-distributions. –  Eckhard Mar 28 '13 at 11:23
(1) is true. It is a consequence of Cochran's theorem. –  Siméon Mar 28 '13 at 11:37
@Eckhard : indeed. I found all I needed on the wikipedia page on the Student distribution. Many thanks. –  Ewan Delanoy Mar 28 '13 at 12:26