Is it possible to calculate the expectation of $\frac{a}{(b+x)^2}$ where x is gamma distributed?

Question

Is it possible to calculate the expectation of $\frac{a}{(b+x)^2}$ where x is gamma distributed?

Or more generally, can you calculate the distribution of $\frac{Na}{(b+\sum\limits_{i=1}^n x_i)^2}$ where the x's are IID gamma distributions?

Sorry for not giving any context, it would be tough to do so without confusing the main question at hand.

Thank you!

Answer 1

If $b \ge 0$, then the transformation $Y = g(X) = a/(b+X)^2$ is monotone for $X > 0$; therefore, the density of $Y$ is easily found via the formula $$f_Y(y) = f_X(g^{-1}(y))\left|\frac{dg^{-1}}{dy}\right| = f_X((a/y)^{1/2} - b) \frac{\sqrt{a}}{2}y^{-3/2},$$ where $f_X(x)$ is a gamma density (parameters not specified). Note that the support of $Y$ is $0 < Y < a/b^2$.

The expected value, however, is more efficiently calculated via the relationship $${\rm E}[g(X)] = \int_{x = 0}^\infty g(x) f_X(x) \, dx;$$ where there is no requirement of monotonicity of $g$. Unfortunately, this integral does not have an elementary closed form for general parameters. If $b < 0$, then the calculation of the density of $Y$ is sightly more complicated, but without further details, I have chosen to omit this part of the discussion for the sake of brevity.

Answer 2

If you have access to a computer algebra system, it is quite easy to solve. For your problem, random variable $X \sim Gamma(\alpha, \beta)$ has pdf say $f(x)$:

Then, you seek the expectation:

where the Expect function is from the mathStatica package for Mathematica.

Notes

1. As disclosure, I should add that I am one of the authors of mathStatica.

2. The ExpIntegralE in the output denotes the exponential integral function http://reference.wolfram.com/language/ref/ExpIntegralE.html