(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let X be normally distributed with the paremeters 0 and σ^{2}. Find:

a. E(X^{2})

b. E(aX^{2}+b)

2. Relevant 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]

3. 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]

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# Homework Help: E(X) and Var(X) for Normal Dist.

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