Consider a discrete random variable X taking values k = 0, 1, 2,... with probabilities where µ > 0. This is the Poisson distribution with parameter µ. We will learn more about this distribution in...

Consider a discrete random variable X taking values k = 0, 1, 2,... with probabilities

where µ > 0. This is the Poisson distribution with parameter µ. We will learn more about this distribution in Chapter 12. This exercise illustrates that the sum of independent Poisson variables again has a Poisson distribution.

a. Let X and Y be independent random variables, each having a Poisson distribution with µ = 1. Show that for k = 0, 1, 2,...

b. Let X and Y be independent random variables, each having a Poisson distribution with parameters λ and µ. Show that for k = 0, 1, 2,...