- #1
PMRDK
- 3
- 0
Homework Statement
Let [itex] g(x) = \frac{1}{x+c}[/itex], where [itex]c[/itex] is a positive constant, and [itex]x [/itex] is a random variable distributed according to the Gamma distribution
[itex]x\sim f(x)=\frac{1}{\Gamma(\alpha) \beta^\alpha} x^{\alpha \,-\, 1} e^{-\frac{x}{\beta}}[/itex].
I wish to calculate the expected value of [itex]g(x)[/itex] with respect to the probability density function [itex]f(x)[/itex].
Homework Equations
The expected value can be calculated as
[tex]E(g(x))=\int_0^∞g(x)f(x)dx = \int_0^∞ \frac{1}{x+c} \frac{1}{\Gamma(\alpha) \beta^\alpha} x^{\alpha \,-\, 1} e^{-\frac{x}{\beta}}dx
[/tex]
The Attempt at a Solution
I have problems with calculating the integral. If [itex]g(x)=\frac{1}{x}[/itex], then the integral would not be too difficult. But the constant in the denominator gives me problems. I have attempted with variable substitutions and integration by parts. However, I have not been able to come up with a solution.
Actually, this is not a homework problem. I posted it here since it is `homework style', so I do not know if it is event possible to calculate the expectation.
Any help is much appreciated.