Sum of iid random variables with an odd distribution I have $G_1,G_2$, iid with probability distribution function $f(y) = Ce^{-y}y^{-1/2}$ where c is a normalizing constant. I am trying to find the distribution of $G_1+G_2$. I have tried transforming and using the jacobian, but I always get something unmanageable. Can anyone think of a good transformation, or a hint to get me started?
Thanks.
 A: Note that the density is undefined if $y \le 0$, so the support of $G_1, G_2$ is on $(0, \infty)$.  The distribution of each of $G_1, G_2$ is Gamma with parameters $\alpha = 1/2$, $\beta = 1$; thus their sum is also Gamma but with parameters $\alpha^* = 2\alpha = 1$ and $\beta = 1$; i.e., exponential with rate $1$.  This is a consequence of the fact that the sum of independent gamma variables $$X_i \sim \operatorname{Gamma}(\alpha_i, \beta)$$ is $$S = \sum_{i=1}^n X_i \sim \operatorname{Gamma}\left(\alpha^* = \sum_{i=1}^n \alpha_i, \beta \right).$$  We can prove this easily using MGFs; but in your case, that might be overkill, since it is easy to see that the PDF of the sum is $$f_S(s) = \int_{y=0}^s f_{G_1}(y) f_{G_2}(s-y) \, dy = \int_{y=0}^s C^2 \frac{e^{-y}}{\sqrt{y}} \frac{e^{-(s-y)}}{\sqrt{s-y}} \, dy \propto e^{-s} \int_{y=0}^s \frac{1}{\sqrt{y(s-y)}} \, dy.$$  Now with an appropriate scaling transformation $$u = y/s, \quad dy = s \, du,$$ we find that the $s$ cancels out to give an integral that is independent of $s$, hence $f_S(s) = e^{-s}$.
