R.v. $Y_i$ is i.i.d. and normally distributed $N(0,\sigma^2)$ for all $i$. Prove that $$E\left(\frac{Y^2}{\sigma^2}\right) = 1$$ and $$W = \frac{1}{\sigma^2}\sum_{i=1}^n Y_i^2$$ is distributed $\chi_n^2$


As for $W \thicksim \chi^2(n)$, corollary to the Hint 2: If $Z_1,Z_2,...,Z_n$ are independent normal random variables with different means and variances, that is: $Z_i \thicksim N(\mu_i,\sigma_I^2)$ for $I = 1,2,...,n.$ Given that $W = \sum_{i=1}^{n} \frac{Y_i^2}{\sigma^2} = \sum_{i=1}^{n} Z_i^2$. Therefore, $W \thicksim X^2(n)$. I hope what I did is right.


closed as off-topic by Did, Juniven, Bram28, Shailesh, астон вілла олоф мэллбэрг Mar 15 '17 at 3:33

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  • $\begingroup$ How far did you get before posting this? What have you tried? $\endgroup$ – Sanderr Mar 14 '17 at 14:46
  • $\begingroup$ Can you determine the law of $Y_i/\sigma$? $\endgroup$ – Augustin Mar 14 '17 at 14:48

Hint 1: $\mathsf{Var}(Y_i) = \mathsf{E}(Y_i^2) - \mathsf{E}(Y_i)^2$

Hint 2: If $X_1, X_2, \cdots, X_k$ are i.i.d. standard normal random variables, then $\sum_i X_i^2 \sim \chi_k^2$.

  • $\begingroup$ How do I use the Hint 2? I used the Hint 1 but it got me nowhere...all I know is that $E(Y^2) = \sigma^2 + \mu^2$ $\endgroup$ – Zander Assand Mar 14 '17 at 16:23
  • $\begingroup$ standard normal r.v. is $Z = \frac{X-\mu}{\sigma}$ $\endgroup$ – Zander Assand Mar 14 '17 at 16:30
  • $\begingroup$ ok so $E(\frac{Y^2}{\sigma^2}) = E(\frac{Y}{\sigma})^2 = E(Z)^2$? $\endgroup$ – Zander Assand Mar 14 '17 at 16:38
  • $\begingroup$ In that case, $E(Z)^2 \thicksim \chi^2(1) = 1$ in which 1 stands for $r$ or degrees of freedom of 1 $\endgroup$ – Zander Assand Mar 14 '17 at 16:45
  • $\begingroup$ I cant see the use of Hint 2 from what I did. I only used a theorem that states that If $X \thicksim \chi^2(r)$, then $E[X] = r$. $\endgroup$ – Zander Assand Mar 14 '17 at 17:00

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