Prove that, for a Poisson arrival process where n arrivals over a fixed time period are independent and identically distributed (i.i.d) with expectation

λ, that, in the limit of large n the probability of k arrivals over the same period is given by:

Demonstrate that the moment generating function for this process is given by:

By differentiation, use the previous expression to demonstrate that both the mean and variance of the Poisson distribution are given by λ. (We will be using these results in the following chapter).

Transcribed Image Text:

ayt P|(X = k) = k!