# Homework Help: Probability density functions

1. Apr 16, 2010

### clockworks

1. The problem statement, all variables and given/known data

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

2. Relevant 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

3. The attempt at a solution

i found &=2/b and thought my solution was coherent but seeing as i cant answer the next question im confused as to where i went wrong .
i manage to find X= a + sqrt(pi/4) but cant 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)

2. Apr 17, 2010

### vela

Staff Emeritus
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.

3. Apr 17, 2010

### clockworks

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

4. Apr 17, 2010

### clockworks

also the probability density function is a Rayleigh distribution

5. Apr 17, 2010

### 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

6. Apr 17, 2010

### clockworks

thx vela problem solved :D