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(a) A hen lays N eggs where N is Poisson with parameter λ. The weight of the nth egg is Wn, where W₁, W2,... are independent identically distributed variables with common probability generating function G(s). Show that the generating function Gw of the total weight W = Σ₁ Wi is given by Gw (s) = exp{-λ + AG (s)). W is said to have a compound Poisson distribution. Show further that, for any positive integral value of n, Gw (s)1/n is the probability generating function of some random variable; W (or its distribution) is said to be infinitely divisible in this regard. (b) Show that if H (s) is the probability generating function of some infinitely divisible distribution on the non-negative integers then H(s) = exp{-λ + AG (s)} for some λ (> 0) and some probability generating function G(s).

(a) A hen lays N eggs where N is Poisson with parameter λ. The weight of the nth egg is Wn, where W₁, W2,... are independent identically distributed variables with common probability generating function G(s). Show that the generating function Gw of the total weight W = Σ₁ Wi is given by Gw (s) = exp{-λ + AG (s)). W is said to have a compound Poisson distribution. Show further that, for any positive integral value of n, Gw (s)1/n is the probability generating function of some random variable; W (or its distribution) is said to be infinitely divisible in this regard. (b) Show that if H (s) is the probability generating function of some infinitely divisible distribution on the non-negative integers then H(s) = exp{-λ + AG (s)} for some λ (> 0) and some probability generating function G(s).

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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(a) A hen lays N eggs where N is Poisson with parameter λ. The weight of the nth egg is Wn, where W₁, W2,... are independent identically distributed variables with common probability generating function G(s). Show that the generating function Gw of the total weight W = Σ₁ Wi is given by Gw (s) = exp{-λ + AG (s)}. W is said to have a compound Poisson distribution. Show further that, for any positive integral value of n, Gw (s)1/n is the probability generating function of some random variable; W (or its distribution) is said to be infinitely divisible in this regard. (b) Show that if H (s) is the probability generating function of some infinitely divisible distribution on the non-negative integers then H(s) = exp{-λ + AG (s)} for some λ (> 0) and some probability generating function G(s).
Transcribed Image Text:(a) A hen lays N eggs where N is Poisson with parameter λ. The weight of the nth egg is Wn, where W₁, W2,... are independent identically distributed variables with common probability generating function G(s). Show that the generating function Gw of the total weight W = Σ₁ Wi is given by Gw (s) = exp{-λ + AG (s)}. W is said to have a compound Poisson distribution. Show further that, for any positive integral value of n, Gw (s)1/n is the probability generating function of some random variable; W (or its distribution) is said to be infinitely divisible in this regard. (b) Show that if H (s) is the probability generating function of some infinitely divisible distribution on the non-negative integers then H(s) = exp{-λ + AG (s)} for some λ (> 0) and some probability generating function G(s).
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