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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.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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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