Not sure how the coefficient in this chi-squared problem affects the expectation and variance? I have written out my answer so far but I'm not sure how the coefficient's might affect the answer. Can anyone help me understand if and what I am doing wrong?
$\sigma^2 = 12; \mu= 170; n= 21$
Using $(n-1)S^2/\sigma^2$ I have that the p.d.f. follows a chi-squared distribution of $20S^2/144$ with 20 degrees of freedom.
$E(20S^2/144) = \sigma^2 = 12^2 = 144$
$Var(20S^2/144) = 2n = 2 * 21 = 42$
Set
$$Y=\frac{(n-1)S^2}{\sigma^2}\sim \chi_{20}^2$$
You have that
$$\mathbb{E}[Y]=n=20$$
and
$$\mathbb{V}[Y]=2n=40$$
First proof (sketch)
just observe that
$$\chi_{n}^2=Gamma\Bigg[\frac{n}{2};\frac{1}{2}\Bigg]$$
and you can easy calculate mean and variance with the integral. Here $1/2$ is the "rate parameter"
Second proof:
Observe also that
$$Y=\frac{(n-1)S^2}{\sigma^2}=\frac{20S^2}{12}$$
Thus
$$\mathbb{E}\Bigg[\frac{20S^2}{12}\Bigg]=\frac{20}{12}\mathbb{E}[S^2]=\frac{20}{12}\sigma^2=20$$
and similarly for its variance