Expectation of Chi square distribution

Question

Given that $Y$ is a degree $8$ chi square distribution i.e $Y \sim \chi_{8}^2$

I want to find $E[Y^2]$ and $Var[Y^2]$

I only know that $E[Y]$ of it should be $8$, but I don't know how to find $E[Y^2]$ from it?

Thank you

Answer 1

$$\chi_{8}^2=Gamma(4;1/2)$$

Where $1/2$ is the rate parameter. Now it is not difficult to calculate any moment you want solving the integrals

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Alternative method:

Given that $Y\sim \chi_8^2$,

$$Y=Z_1^2+\dots +Z_8^2$$

Where $Z_i$ are iid Standard gaussian. Moments of the standard gaussian are well known thus...

$$\mathbb{E}[Y]=\underbrace{1+1+\dots +1}_{\text{8 times}}=8$$

$$\mathbb{V}[Y]=8\Bigg[\frac{4!}{2^2\cdot 2!}-1\Bigg]=16$$

Answer 2

$X\sim \chi_{n}^2$

$E(X)=n,V(X)=2n$

$E(X^2)=V(X)+E(X)^2$

Above mentioned stuff is sufficient to solve this problem