A couple questions with chi square

**Janu42** · March 28th 2010, 07:26 PM

1) Use the fact that $(n-1)^2$ $S^2$ / $\sigma^2$ is a chi square random variable with n-1 degrees of freedom to prove that
$Var(S^2)$ = $2\sigma^4/n-1$ .

Hint: Use the fact that the variance of a chi square random variable with k degrees of freedom is 2k.

2) Let V and U be independent chi square random variables with 7 and 9 degrees of freedom, respectively. Is it more likely that (V/7)/(U/9) will be between
a) 2.51 and 3.29 or
b) 3.29 and 4.20

**matheagle** · March 28th 2010, 08:15 PM

The square on (n-1) in your statement is incorrrect....

Originally Posted by Janu42

1) Use the fact that ${(n-1)S^2\over \sigma^2}$ is a chi square random variable with n-1 degrees of freedom to prove that
$Var(S^2)$ = $2\sigma^4/n-1$ .

Hint: Use the fact that the variance of a chi square random variable with k degrees of freedom is 2k.

2) Let V and U be independent chi square random variables with 7 and 9 degrees of freedom, respectively. Is it more likely that (V/7)/(U/9) will be between
a) 2.51 and 3.29 or
b) 3.29 and 4.20

$V(S^2) = V\left (S^2{ (n-1)\sigma^2\over (n-1)\sigma^2}\right)$

$= \left({\sigma^4\over (n-1)^2}\right) V\left (S^2{ (n-1)\over \sigma^2}\right)$

$= \left({\sigma^4\over (n-1)^2}\right) V\left (\chi^2_{n-1}\right)$

$= \left({\sigma^4\over (n-1)^2}\right) 2(n-1)$

(b) Use an F distribution table

**Janu42** · March 28th 2010, 08:26 PM

You're right, sorry i typoed. Thanks.

For #2, do I use the pdf for a chi square?

EDIT: Sorry, typed that while you were editing yours I guess. But OK, thanks a lot.

**matheagle** · March 28th 2010, 08:29 PM

To be precise $F_{7,9}$ and look up those numbers.
Remember the mean of an F is near 1, so you know which tail to use in this case.

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