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Let X1, X₂,... be independent variables each taking the values 0 or 1 with probabilities 1 - p and p, where 0 < p < 1. Let N be a random variable taking values in the positive integers, independent of the X₁, and write S = X₁ + X₂ + + XN. Write down the conditional generating function of N given that S = N, in terms of the probability generating function G of N. Show that N has a Poisson distribution if and only if E(x)P = E(x | S = N) for all p and x.

Let X1, X₂,... be independent variables each taking the values 0 or 1 with probabilities 1 - p and p, where 0 < p < 1. Let N be a random variable taking values in the positive integers, independent of the X₁, and write S = X₁ + X₂ + + XN. Write down the conditional generating function of N given that S = N, in terms of the probability generating function G of N. Show that N has a Poisson distribution if and only if E(x)P = E(x | S = N) for all p and x.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.CR: Chapter 12 Review
Problem 84CR
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Let X1, X₂,... be independent variables each taking the values 0 or 1 with probabilities 1 - p and p, where 0 < p < 1. Let N be a random variable taking values in the positive integers, independent of the X₁, and write S = X₁ + X₂ + + XN. Write down the conditional generating function of N given that S = N, in terms of the probability generating function G of N. Show that N has a Poisson distribution if and only if E(x)P = E(xN | S = N) for all p and x.
Transcribed Image Text:Let X1, X₂,... be independent variables each taking the values 0 or 1 with probabilities 1 - p and p, where 0 < p < 1. Let N be a random variable taking values in the positive integers, independent of the X₁, and write S = X₁ + X₂ + + XN. Write down the conditional generating function of N given that S = N, in terms of the probability generating function G of N. Show that N has a Poisson distribution if and only if E(x)P = E(xN | S = N) for all p and x.
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