integration of a pdf, expected value

alias

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.

alias

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

Ray Vickson

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

alias

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

Ray Vickson

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

RGV

Robert1986

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?

alias

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.