Concept explainers
Each customer making a particular Internet purchase must pay with one of three types of credit cards (think Visa, MasterCard, AmEx). Let Ai (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 Xi = 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.]
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Probability and Statistics for Engineering and the Sciences
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,