In this exercise, we shall prove that the three assumptions underlying the Poisson process model do indeed imply that occurrences happen according to a Poisson process. What we need to show is that, for each t, the number of occurrences during a time interval of length t has the Poisson distribution with mean λt. Let X stand for the number of occurrences during a particular time interval of length t. Feel free to use the following extension of Eq. (5.4.7): For all real a,

a. For each positive integer n, divide the time interval into n disjoint subintervals of length t/n each. For i = 1, . . . , n, let Yi = 1 if exactly one arrival occurs in the ith subinterval, and let Ai be the event that two or more occurrences occur during the ith subinterval.
Let

For each nonnegative integer k, show that we can write Pr(X = k) = Pr(Wn = k) + Pr(B), where B is a subset of

b. Show that limn†’ˆž Pr(ˆªni=1Ai) = 0. Show that

c. Show that limn†’ˆž Pr(Wn = k) = eˆ’λ(λt)k/k!. limn†’ˆž n!/[nk(n ˆ’ k)!] = 1.
d. Show that X has the Poisson distribution with mean λt.

lim 1+au()e(5.4.9) i-1A Pr(nn_ 149 = (1 + o(u))1/u where u = 1/n