Question

4. Assume that the number of car accidents on Saturday has a Poisson dis- tribution with mean 3.7. Assume that the number of car accidents on Sunday has a Poisson distribution with mean 2.3. Assume that the two random variables are independent. What is the distribution of their sum?

Answer 1

Let, X and Y denote car accidents on Saturday and Sunday respectively, so as per the problem,

$X \sim Poisson (\lambda)$

$Y \sim Poisson (\mu)$

where, $\lambda$ and $\mu$ are the mean of the poisson distribution X and Y are following respectively.

So it is proved that,

$X + Y \sim Poisson(\lambda + \mu)$

So the sum will of the poisson distribution will follow poisson distribution with mean $\lambda + \mu$

Given the problem, $\lambda$ = 3.7 and $\mu$ = 2.3

So the sum of two independent poisson distribution is another poisson distribution with mean = 3.7 + 2.3 = 6.