- #1

- 780

- 4

## Homework Statement

Let X be normally distributed with the paremeters 0 and σ

^{2}. Find:

a. E(X

^{2})

b. E(aX

^{2}+b)

## Homework Equations

E(X) = [itex] \int_{-\infty}^{\infty} \! xf(x) \mathrm{d} x [/itex]

E(X

^{2}) = [itex] \int_{-\infty}^{\infty} \! x^2f(x) \mathrm{d} x [/itex]

E(aX+b) = [itex] aE(X)+b [/itex]

The normal distribution with paremeters 0 and σ

^{2}= [itex]( \frac{1}{\sigma \sqrt{2 \pi }}) e^{ \frac{-x^2}{2 \sigma ^2}} [/itex]

## The Attempt at a Solution

a. [itex] E(X^2)= \frac{1}{ \sigma \sqrt{2 \pi }} \int_{-\infty}^{\infty} \! x^2e^{ \frac{-x^2}{2 \sigma ^2}} \mathrm{d} x [/itex]

I don't knowhow to solve this integral. I think there might some tricks involved using the fact that this is a distribution function.

b. I just need to find E(X) which I start by setting up: [itex] E(X)= \frac{1}{ \sigma \sqrt{2 \pi }} \int_{-\infty}^{\infty} \! xe^{ \frac{-x^2}{2 \sigma ^2}} \mathrm{d} x [/itex]

I don't know how to solve this integral either.

I also don't know how to solve the simpler integral [itex] \int e^{ \frac{-x^2}{2 \sigma ^2}} \mathrm{d} x [/itex]