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 $\Gamma(n)=(n-1)!$.

  • 2
    $\begingroup$ @Ryan You could also compare characteristic functions. $\endgroup$
    – drhab
    Aug 28 '15 at 20:07
  • $\begingroup$ @drhab How would that help in the calculation? $\endgroup$
    – Kerry
    Aug 28 '15 at 21:32
  • $\begingroup$ @Augustin - got it thanks. $\endgroup$
    – Kerry
    Aug 28 '15 at 22:20
  • 1
    $\begingroup$ @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. $\endgroup$
    – drhab
    Aug 29 '15 at 9:26
  • $\begingroup$ 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. $\endgroup$
    – Augustin
    Aug 29 '15 at 9:31

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