Let the random variable X Poi() describe the number of people that enter an elevator on the ground floor of a building with N

Let the random variable X ∼ Poi(λ) describe the number of people that enter an elevator on the ground floor of a building with N + 1 floors. Let Y be the number of stops that the elevator needs to make, assuming that the X people independently choose one of the N floors and each floor is equally likely to be chosen.

(a) Show that E [Y ] = N 1 − e−λ/N

. (Hint: for i = 1,...,N, define

so that Y = N i=1 Ii.)

(b) Interpret the result for N = 1 and explain what happens as λ increases.

(c) Explain intuitively what limN→∞ E [Y ] should be.

(d) Prove your result from the previous question. (Hint: either try a direct Taylor series approach, or consider the substitution M = 1.)

1, if the elevator stops on floor i, 10, otherwise,