1. ## Proving of Distribution

Hi all,

I was reading through my textbook and I could not solve any of these questions. The painful part is, I don't even know how and where to start. It'll be great if anyone of you could help me out. I've spend like hours looking through the textbook and internet for solution but to no avail.

I presume that you know that "$\displaystyle N(0,\sigma^2)$", the normal distribution with mean 0 and standard deviation $\displaystyle \sigma$, has density function $\displaystyle \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{x^2}{2\sigma^2}}$. The "expected value", E(f(x)), for any function f, is $\displaystyle \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^\infty f)x) e^{-\frac{x^2}{2\sigma^2}}$. With $\displaystyle f(x)= \frac{x^2}{\sigma^2}$, that is $\displaystyle \frac{1}{\sigma^3\sqrt{2\pi}}\int_{-\infty}^\infty x^2 e^{-\frac{x^2}{2\sigma^2}}$. Do that integral using "integration by parts", twice.