Let X be a discrete random variable taking nonnegative

Chapter , Problem 38

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Let X be a discrete random variable taking nonnegative integer values. Let M(s) be the transform associated with X. (a) Show that P(X = 0) = lim M(s). s --oo (b) Use part (a) to verify that if X is a binomial random variable with parameters n and p, we have P(X = 0) = (1 - pt. Furthermore, if X is a Poisson random variable with parameter .A. we have P( X = 0) = e - A . (c) Suppose that X is instead known to take only integer values that are greater than or equal to a given integer k. How can we calculate P(X = k) using the transform associated with X? Solution. (a) We have oc Al(s) = L P(X = k)ek s. k=O As s ---+ -oc, all the terms e k s with k > 0 tend to 0, so we obtain lims_-::>e A1(s) = P(X = 0). (b) In the case of the binomial, we have from the transform tables so that lims_-oc Al(s) = (1 - p)n . In the case of the Poisson, we have so that lims_-oc Al(s) = e- A (c) The random variable Y = X - k takes only nonnegative integer values and the associated transform is A1y (s) = e-SkAl(s) (cf. Example 4.25). Since P(Y = 0) = P(X = k), we have from part (a), P(X = k) = lim esk Al(s). s-oo