Determine the distribution of the sum of n independent identically distrubted poisson random variable $X_i$?

Question

Determine the distribution of the sum of n independent identically distributed Poisson random variable $X_i$ ~ $\mathsf{Poisson}(\lambda)$, $ i = 1, ...., n$?

My approach is that since it is independent identically and the sum is basically $X_1 + X_2 + ... + X_n$ we can just use moment generate function to find it.

So $M_x (t) = E e^{xt} = e^{-\lambda} * e^{e^{\lambda*t} },$ which is just basically that to the $n$ power.

Can anyone confirm that I'm on the right path?

Answer 1

Your technique will work, and is a good exercise to help learn to work with generating functions.

The easy way to do the problem, however, is to note that a Poisson distribution represents the number of events in a fixed interval of time, given that the events occur with a given fixed average rate. The rate, in events per second, of the sum of these will be the sum of the rates. Thus if all the independent $X_i$ have $\lambda = \lambda_1$ the answer will be a Poisson distribution with $$ \lambda = \sum \lambda_i = n\lambda_1 $$

Answer 2

Yes. MGFs are a convenient way to show that the sum of independent Poisson random variables is again Poisson. The rate (mean) $\lambda_S$ for the sum is just the sum of the rates of the individual $X_i \sim Pois(\lambda_i)$ being summed.

Here is a brief simulation in R statistical software for three independent Poisson random variables $X_i \sim Pois(\lambda_i),$ with $\lambda_1 = 1,$ $\lambda_2 = 3,$ and $\lambda_3 = 6.$ Then $S = X_1 + X_2 + X_3 \sim Pois(\lambda_S = 10).$ With a million realizations of the $X_i$ and $S$, expectations and variances should be accurate to at least two places.

m = 10^6
x1 = rpois(m,1);  x2 = rpois(m,3);  x3 = rpois(m, 6)
mean(x1);  mean(x2);  mean(x3)  # expectations near 1, 3, 6
## 0.998909
## 2.999257
## 6.002289
var(x1);  var(x2);  var(x3)     # variances near 1, 3, 6
## 0.9969768
## 3.003303
## 6.001134
s = x1 + x2 + x3
mean(s);  var(s)                # aprx E(s) = 10,  V(S) = 10
## 10.00046
## 10.01897

Below is a histogram of the one million simulated realizations of $S.$ The red dots atop histogram bars show exact values of $Pois(10).$