Finding the Expected Value of X with a Probability Density Function

[clockworks]

Homework Statement

f(x)=&(x-a)exp((-(x-a)^2)/b) where a and b are constants

Homework Equations

find & in terms of b:

show that the expected value of X is given by
X=a + sqrt(pi*b/4)
identity given
x(x-a)=(x-a)^2+a(x-a)
and integral from 0 to infinity of x^2*exp-x^2 dx=sqrt (pi) /4

The Attempt at a Solution

i found &=2/b and thought my solution was coherent but seeing as i can't answer the next question I am confused as to where i went wrong .
i manage to find X= a + sqrt(pi/4) but can't get that b into the square root no matter what i try .
i separated into 2 integrals using the first identity then set Y=(x-a)/sqrt b and used the second identity to get sqrt (pi /4)( the other integral giving the expected a)

 

[vela]

Did you remember to write dx in terms of dy when you did the substitution?

By the way, it would help in the future if you provide the complete problem statement. You didn't tell us what the domain of f(x) was, for instance.

 

[clockworks]

im sorry the domain of fx is the function provided for x>=a and 0 otherwise

 

[clockworks]

also the probability density function is a Rayleigh distribution

 

[clockworks]

using wikipedia i found the correct answer (using the formulas that use the variance and such) but id still like to know how to recalculate it using the identities given so my question still stands :D

 

[clockworks]

thx vela problem solved :D