ArcanaNoir
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Homework Statement
Let X be normally distributed with the paremeters 0 and σ2. Find:
a. E(X2)
b. E(aX2+b)
Homework Equations
E(X) = \int_{-\infty}^{\infty} \! xf(x) \mathrm{d} x
E(X2) = \int_{-\infty}^{\infty} \! x^2f(x) \mathrm{d} x
E(aX+b) = aE(X)+b
The normal distribution with paremeters 0 and σ2 = ( \frac{1}{\sigma \sqrt{2 \pi }}) e^{ \frac{-x^2}{2 \sigma ^2}}
The Attempt at a Solution
a. E(X^2)= \frac{1}{ \sigma \sqrt{2 \pi }} \int_{-\infty}^{\infty} \! x^2e^{ \frac{-x^2}{2 \sigma ^2}} \mathrm{d} x
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: E(X)= \frac{1}{ \sigma \sqrt{2 \pi }} \int_{-\infty}^{\infty} \! xe^{ \frac{-x^2}{2 \sigma ^2}} \mathrm{d} x
I don't know how to solve this integral either.
I also don't know how to solve the simpler integral \int e^{ \frac{-x^2}{2 \sigma ^2}} \mathrm{d} x