# Thread: integration to find expectation

1. ## integration to find expectation

Hi

I need to work out this integral $\displaystyle $$\int^{\infty}_{0}a^2y^2e^{\frac{-a^2y^2}{2}}dy$$$. I have already multiplied the expression by y, since i'm trying to work out E(Y).

Now I tried to look at the Gamma and Beta functions, but I can't recognise how they would work here, mainly because of the exponential part contains a $\displaystyle y^2$.

Any help is much appreciated

2. ## Re: integration to find expectation

Rewrite this as 1/2 the expected value of $y^2$ when $y$ is zero mean normally distributed with standard deviation $\dfrac 1 a$

You'll probably also have to multiply by a constant to the get the proper form.

3. ## Re: integration to find expectation

Hi Romsek

my PDF was actually this $\displaystyle $$\int^{\infty}_{0}a^2ye^{\frac{-a^2y^2}{2}}dy$$$

hence i wrote $\displaystyle $$\int^{\infty}_{0}a^2y^2e^{\frac{-a^2y^2}{2}}dy$$$ because I wanted E(Y).

4. ## Re: integration to find expectation

Originally Posted by cooltowns
Hi Romsek

my PDF was actually this $\displaystyle $$\int^{\infty}_{0}a^2ye^{\frac{-a^2y^2}{2}}dy$$$

hence i wrote $\displaystyle $$\int^{\infty}_{0}a^2y^2e^{\frac{-a^2y^2}{2}}dy$$$ because I wanted E(Y).
Whatever. The method I suggested will allow you to compute the integral.