• Support PF! Buy your school textbooks, materials and every day products Here!

Integration of a pdf, expected value

  • Thread starter alias
  • Start date
  • #1
46
0

Homework Statement


Show that E[Y^4] = 3, where Y~N(0,1)

Homework Equations


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

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.
 

Answers and Replies

  • #2
46
0
Sorry, it's a definite integral in from -inf to inf.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

Homework Statement


Show that E[Y^4] = 3, where Y~N(0,1)

Homework Equations


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

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.
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
46
0
I tried integration by parts but the integral I end up with makes no sense to me.
 
  • #5
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
I tried integration by parts but the integral I end up with makes no sense to me.
If you show your work we have the basis of a discussion.

RGV
 
  • #6
828
2
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
46
0
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.
 

Related Threads on Integration of a pdf, expected value

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
5K
Replies
2
Views
855
Replies
5
Views
2K
Replies
1
Views
1K
Replies
5
Views
5K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
1
Views
4K
Top