Suppose that the number of events occurring in a given time period is a Poisson random variable with parameter λ. If each event is classified as a type i event with probability pi, i = 1, . . . , n, ∑ pi = 1, independently of other events, show that the numbers of type i events that occur, i = 1, . . . , n, are independent Poisson random variables with respective parameters λpi, i = 1, . . . , n.
Answer to relevant QuestionsLet U denote a random variable uniformly distributed over (0, 1). Compute the conditional distribution of U given that (a) U > a; (b) U < a; where 0 < a < 1. Suppose that F(x) is a cumulative distribution function. Show that (a) Fn(x) and (b) 1 − [1 − F(x)]n are also cumulative distribution functions when n is a positive integer. Let X1, . . . ,Xn be independent random ...If X and Y are independent standard normal random variables, determine the joint density function of U = X V= X/Y Then use your result to show that X/Y has a Cauchy distribution. Let X be a normal random variable with mean μ and variance σ2. Use the results of Theoretical Exercise 46 to show that In the preceding equation, [n/2] is the largest integer less than or equal to n/2. Check your answer by ...A group of n men and n women is lined up at random. (a) Find the expected number of men who have a woman next to them. (b) Repeat part (a), but now assuming that the group is randomly seated at a round table.
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