# Integration of a pdf, expected value

1. ### alias

46
1. The problem statement, all variables and given/known data
Show that E[Y^4] = 3, where Y~N(0,1)

2. Relevant equations
E[(Y-mu)^4]/[E(Y-mu)^2]^2 = 3
E(Y^4) = 1/sprt(2pi)*intregral (y^4)*e^(-y^2/2)

3. The attempt at a solution
I have expanded and simplified the first equation above and cannot get it to equal 3. I think it's possible to solve the second using integration by parts but I can't find what to use for
[u, du, dv, v] in order to integrate by parts. Any help would be appreciated. Thanks.

2. ### alias

46
Sorry, it's a definite integral in from -inf to inf.

3. ### Ray Vickson

6,010
You might have to try several things until you get something that will work. What have you tried so far? Another approach that is often useful is to look at dI(a)/da or d^2I(a)/da^2, where I(a) = integral_{y=-inf..inf} exp(-a*y^2/2) dy.

RGV

4. ### alias

46
I tried integration by parts but the integral I end up with makes no sense to me.

5. ### Ray Vickson

6,010
If you show your work we have the basis of a discussion.

RGV

6. ### Robert1986

828
Just spit-balling, is there a way to use some substitutions to make this look something like the pdf of a gamma distribution? Then use what you know about that?

7. ### alias

46
I integrated the Gaussian distribution, it took a long time but I finally got the right answer. After making a substitution, integration by parts worked. I would like to know if the formula: E[(Y-mu)^4]/[E(Y-mu)^2]^2, is useless for answering this question though. Thanks guys.

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