# Expected value of Y = (1/X) where $X \sim Gamma$

I'm having some confusion over this statement here. Let $T_i \sim Exp(\lambda + \theta)$ and if they are all iid then $\sum_n T_i \sim Gamma(\alpha = n, \beta = 1/(\lambda + \theta))$

I want to find $E(\frac{1}{\sum_n T_i})$. I know I can't just do the reciprocal of $E(\sum_n T_i)$

Do I integrate over $\int_0^\infty \frac{t}{f(t)} dt$?

If I do this I get $\Gamma(\alpha) \beta^\alpha \int_0^\infty \frac{e^{t/\beta}}{t^\alpha} dt$ and I don't know if this is the right way to proceed since I end up with incomplete gamma?

I know that I'm supposed to end up with $\frac{\lambda + \theta}{n-1}$ as the answer but I'm not sure how the gamma is supposed to cancel out.

• the density goes in the numerator...... – P.Windridge Feb 26 '15 at 18:19
• and I think you mean the answer should be $(\lambda + \theta)/(n-1)$ – P.Windridge Feb 26 '15 at 18:27
• The density should go in the numerator but would it still be the density function I'm integrating over if I want 1/X and not X? And yes you are correct about the answer, I'll fix that typo. Thanks – miruku Feb 26 '15 at 18:28
• If $X$ is a random variable with density $f$, then for any (nice) function $G$, $\mathbb E[ G(X) ] = \int G(x)f(x)dx$ – P.Windridge Feb 26 '15 at 18:31
• One easy way to do it is to apply the abovementioned law, and play 'spot the density', pull out the appropriate constants and cross out the integral of the density (since that integrates to 1). – Glen_b Feb 26 '15 at 21:31

The calculation of $E\left[X^{-1}\right]$ when $X$ is a Gamma random variable with order parameter $n$ and rate parameter $\lambda$ requires recognition of the density of another Gamma random variable (with order parameter $n-1$ and rate parameter $\lambda$) in the integral given by the law of the unconscious statistician for $E\left[X^{-1}\right]$. We have
\begin{align} E\left[X^{-1}\right]&= \int_0^\infty \frac 1x \cdot \underbrace{\lambda \frac{(\lambda x)^{n-1}}{\Gamma(n)}e^{-\lambda x}}_{\Gamma(n,\lambda)~\text{density}}\,\mathrm dx\\ &= \lambda\frac{\Gamma(n-1)}{\Gamma(n)}\int_0^\infty \underbrace{\lambda \frac{(\lambda x)^{n-2}}{\Gamma(n-1)}e^{-\lambda x}}_{\Gamma(n-1,\lambda)~\text{density}}\,\mathrm dx\\ &= \frac{\lambda}{n-1} \end{align} since for positive integer $k$, $\Gamma(k) = (k-1)!$.