3. Compounding. (a) Let X have the Poisson distribution with parameter Y, where Y has the Poisson distribution with parameter μ. Show that Gx+y(x) = exp{μ(xe*-1-1)}. (b) Let X₁, X2,... be independent identically distributed random variables with the logarithmic mass function f (k) = (1 - p)k k log(1/p) k≥ 1, where 0 < p < 1. If N is independent of the X; and has the Poisson distribution with parameter μ, show that Y = 1 X; has a negative binomial distribution.
3. Compounding. (a) Let X have the Poisson distribution with parameter Y, where Y has the Poisson distribution with parameter μ. Show that Gx+y(x) = exp{μ(xe*-1-1)}. (b) Let X₁, X2,... be independent identically distributed random variables with the logarithmic mass function f (k) = (1 - p)k k log(1/p) k≥ 1, where 0 < p < 1. If N is independent of the X; and has the Poisson distribution with parameter μ, show that Y = 1 X; has a negative binomial distribution.
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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Transcribed Image Text:3. Compounding.
(a) Let X have the Poisson distribution with parameter Y, where Y has the Poisson distribution with
parameter u. Show that Gx+y(x) = exp{u(xe*-1-1)}.
(b) Let X₁, X2,... be independent identically distributed random variables with the logarithmic
mass function
f(k)=
(1 - p)k
k log(1/p)'
k≥ 1,
where 0 < p < 1. If N is independent of the X; and has the Poisson distribution with parameter
μ, show that Y = ₁ X; has a negative binomial distribution.
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