Question

Let X and Y be independent random variables following Poisson distributions, each with parameter λ = 1. Show that the distribution of Z = X + Y is Poisson with parameter λ = 2. using convolution formula

Answer 1

Probability Mass Function of X Poisson with parameter = 1 is given by:

Probability Mass Function of Y Poisson with parameter = 1 is given by:

To find the distribution of Z = X + Y

By Convolution Theorem:

Multiplying and dividing by z!, we get:

                              (1)

By Binomial Theorem, we have:

                              

i.e.,

Substituting (2), equation (1) becomes:

Thus, we prove that the distribution of Z = X + Y is Poisson with parameter = 2.