Question

Supposex ~N(0,1) and Y=X2 Write down the p.d.f. of Y. Calculate E(Y) and Var(Y).

Answer 1

If the random variable $V \sim N(\mu, \sigma^2),\sigma^2>0,$ then the random variable $W=\dfrac{(X-\mu)^2}{\sigma^2}=Z^2\sim \chi^2(1),$ chi-squared distribution, with $\nu=1$ degrees of freedom.

Suppose $X\sim N(0, 1),$ then $Y=\dfrac{(X-0)^2}{1^2}=X^2\sim \chi^2(1),$ chi-squared distribution, with $\nu=1$ degrees of freedom.

Its density function is given by

$$f(y;\nu)= \begin{cases} \dfrac{1}{2^{\nu/2}\Gamma(\nu/2)}y^{\nu/2-1}e^{-y/2},y>0 \\ 0, \text{elsewhere} \end{cases}$$

Then

$$f(y;\nu)= \begin{cases} \dfrac{1}{\sqrt{2\pi}\sqrt{y}}e^{-y/2},y>0 \\ 0, \text{elsewhere} \end{cases}$$

The mean of the distribution is equal to the number of degrees of freedom:

$$E(Y)=\mu_Y=\nu=1$$

The variance is equal to two times the number of degrees of freedom:

$$Var(Y)=\sigma_Y^2=2\nu=2$$