Independent random variables $Y_i$ with $Y_i \sim N(0,\sigma^2)$

Question

I am struggeling with this problem:

How to show that, $E(Y^2 _i /σ^2) = 1$

I tried:

$$E(Y^2 _i /σ^2)=E(Y^2 _i /(Y_i - \bar{Y})^2) = E(Y^2 _i /(Y^2 _i - 2Y_i\bar{Y} + \bar{Y}))=...$$

However, I am stuck here, because I do not get the transition to $=1$. Pls give me a hint.

Answer 1

$E(Y^2/\sigma^2)=\frac{E(Y^2)}{\sigma^2}=\frac{E(Y-\bar{Y})^2+(E(Y))^2}{\sigma^2}=\frac{\sigma^2+0}{\sigma^2}=1$

Answer 2

Hint: use the linearity of the expectation ($E[aX]=aE[X]$) and the definition of the 2nd moment.

Answer 3

It seems that you're confusing the true variance $\sigma^2$ with something that appears in the usual estimator for the variance. The parameter $\sigma^2$ is nothing else than a fixed positive number (however unknown) and therefore you can just use basic properties of the expectation. For example use that $$ \mathrm{Var}(X)=E[X^2]-E[X]^2,\quad E[c\cdot X]=c\cdot E[X],\;c\in\mathbb{R} $$ for any random variable $X$.