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Let X and Y denote two independent Poisson random variables with and .

Let X and Y denote two independent Poisson random variables with and .

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Exercise 1.6 Let X and Y denote two independent Poisson random variables with parameters and μ. (a) Show that the random variable X + Y has the Poisson distribution with parameter X+μl. (b) Compute the conditional distribution P(X = k | X + Y = n) given that X + Y = n, for all k, ne N. (c) Assume that respective parameters of the distributions of X and Y are random, independent, and chosen according to an exponential distribution with parameter 0 > 0. Give the probability distributions of X and Y, and compute the conditional distribution P(X = k | X + Y = n) given that X + Y = n, for all k, ne N. (d) Assume now that X and Y have same random parameter represented by a single exponentially distributed random variable A with parameter > 0, independent of X and Y. Compute the conditional distribution P(X = k | X + Y = n) given that X + Y = n, for all k, n € N.
Transcribed Image Text:Exercise 1.6 Let X and Y denote two independent Poisson random variables with parameters and μ. (a) Show that the random variable X + Y has the Poisson distribution with parameter X+μl. (b) Compute the conditional distribution P(X = k | X + Y = n) given that X + Y = n, for all k, ne N. (c) Assume that respective parameters of the distributions of X and Y are random, independent, and chosen according to an exponential distribution with parameter 0 > 0. Give the probability distributions of X and Y, and compute the conditional distribution P(X = k | X + Y = n) given that X + Y = n, for all k, ne N. (d) Assume now that X and Y have same random parameter represented by a single exponentially distributed random variable A with parameter > 0, independent of X and Y. Compute the conditional distribution P(X = k | X + Y = n) given that X + Y = n, for all k, n € N.
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