Chapter 5, Problem 88SE

Each customer making a particular Internet purchase must pay with one of three types of credit cards (think Visa, MasterCard, AmEx). Let A_i (i = 1, 2, 3) be the event that a type i credit card is used, with P(A₁) 5 .5, P(A₂) = .3, and P(A₃) = .2. Suppose that the number of customers who make such a purchase on a given day is a Poisson rv with parameter λ. Define rv’s X₁, X₂, X₃ by X_i = the number among the N customers who use a type i card (i = 1, 2, 3). Show that these three rv’s are independent with Poisson distributions having parameters .5 λ, .3 λ, and .2 λ, respectively. [Hint: For non-negative integers x₁, x₂, x₃, let n = x₁ = x₂ = x₃. Then P(X₁ = x₁, X₂= x₂, X₃ = x₃) = P(X₁ = x₁, X₂ = x₂, X₃ = x₃, N = n) [why is this?]. Now condition on N = n, in which case the three Xi’s have a trinomial distribution (multinomial with three categories) with category probabilities .5, .3, and .2.]