# Finding the distribution of the sum of n independent random variables having exponential distributions

My graduate level probability class asks us to calculate the distribution of the sum of n independent exponentially distributed random variables.

I am trying to perform many convolutions but it gets very complicated. Can anyone recommend a better path?

If all the exponential random variables have the same parameter $\lambda$, the answer is $\Gamma(n,\lambda)$ (Gamma distribution), whose density is:
$$\frac{\lambda^n}{\Gamma(n)}e^{-\lambda x}x^{n-1},\, x>0$$
First prove it with $n=2$ using a convolution, this is pretty easy. Then you can prove the general case by induction. Note that $\Gamma(n)=(n-1)!$.
• @Ryan If $X_1,\dots,X_n$ are iid exponential and $S=X_1+\cdots+X_n$ then $\phi_S(t)=(\phi_{X_1}(t))^n$. Find $\phi_Z(t)$ where $Z$ has the distribution mentioned in this answer. It will appear to coincide with function $\phi_S(t)$. Ready, and no calculation of convolutions has been made. Aug 29 '15 at 9:26
• Yes, this is a simple solution if you already know about the gamma distribution and its characteristic function. Otherwise, you have to use the inversion formula on $\phi_Z$ to find the density function. Aug 29 '15 at 9:31