# Distribution of sum of exponentially distributed random variables

By applying the R-function replicate() generate $$10$$ samples from an exponential distribution with a rate parameter $$0.2$$ and sum them together. Do this sum 10000 times and make a histogram of the simulation. Can you say something about the distribution?

So I did this task and observed that the median is $$\approx 50=10/0.2$$ but I don't know what I could say about the distribution. I know that the sum of independent exponentially distributed random variables with same rate is gamma distributed. Is this the answer to the question?

• This distribution is almost exactly normally distributed by the central limit theorem. – Peter Foreman May 8 at 20:13
• The sum of a fixed number $n$ of exponential RVs with the same parameter $a$ follows a classical distribution : gamma distribution $\gamma(n,a)$ known to have the mean you have given. – Jean Marie May 8 at 20:28
• The exponential dist'n is right skewed with a heavy right tail so $n = 10$ is not large enough for the CLT to give useful approximations. By contrast, $m = 10^6$ gives a large enough sample if $S$'s for sample mean $\bar S$ by LLN to approx $\mu_S =E(S)$ and by CLT fot 1.96*sd(s)/dqrt(m) to provide an accurate 95% margin of simulation error for $\mu_S.$ – BruceET May 9 at 15:41

Yes, the sum is $$S = \mathsf{Gamma}(10, \lambda = 0.2).$$ The R code below simulates this distribution with a million iterations, makes a histogram and plots the gamma density for comparison.

set.seed(2019)   # for reproducibility
n = 10; m = 10^6; lam = .2
s = replicate(m, sum(rexp(n, lam)))
mean(s)
 50.00628    # by LLN aprx E(S) = 10/.2 = 50
2*sd(s)/sqrt(m)
 0.03165734  # by CLT aprx 95% margin of sim error

hist(s, prob=T, br=40, col="skyblue2")
curve(dgamma(x, n, lam), 0, max(s), add=T, col="brown", lwd=2) 