Explanation Given information: Variable X is a Poisson random variable with parameter λ . Explanation: A random variable X is a Poisson random variable with parameter λ . Thus, Pr ( X = x ) = λ x x ! e − λ . The probability that X is an even integer (including 0 ) is Pr ( X = 0 ) + Pr ( X = 2 ) + Pr ( X = 4 ) + Pr ( X = 6 ) + .... ⇒ λ 0 0 ! e − λ + λ 2 2 ! e − λ + λ 4 4 ! e − λ + λ 6 6 ! e − λ + ... = e − λ ( λ 0 0 ! + λ 2 2 ! + λ 4 4 ! + λ 6 6 ! + ... ) Thus, the probability that X is an even integer (including 0) is e − λ ( λ 0 0 ! + λ 2 2 ! + λ 4 4 ! + λ 6 6 ! + ... ) To prove the statement that the probability that X is an even integer (including 0) is e − λ cosh λ , shows that e − λ cosh λ = e − λ ( 1 + ( λ 2 2 ! ) + ( λ 4 4 ! ) + ( λ 6 6 ! ) + ... ) . Simplify e − λ cosh λ . From Exercise 23 in Section 11.5 The hyperbolic cosine of x is Cosh x = 1 2 ( e x + e − x ) . The Taylor series expansion of e x is e x = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! +

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Author: Goldstein

Publisher: PEARSON

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Chapter 12.5, Problem 22E

To determine

To prove: The probability that X is an even integer (including 0) is e−λcoshλ, where X

is a Poisson random variable with parameter λ.

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Chapter 12.5, Problem 21E

Chapter 12.5, Problem 23E

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To prove: The probability that X is an even integer (including 0 ) is e − λ cosh λ , where X is a Poisson random variable with parameter λ .

Solution Summary: The author explains that X is a Poisson random variable with parameter lambda and the probability that it is an even integer is

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To prove: The probability that X is an even integer (including 0) is e−λcoshλ, where X

is a Poisson random variable with parameter λ.

Want to see the full answer?

Chapter 12 Solutions

To prove: The probability that X is an even integer (including 0 ) is e − λ cosh λ , where X is a Poisson random variable with parameter λ .

Solution Summary: The author explains that X is a Poisson random variable with parameter lambda and the probability that it is an even integer is

Concept explainers

Videos

To prove: The probability that X is an even integer (including 0) is e−λcoshλ, where X is a Poisson random variable with parameter λ.

Want to see the full answer?

Chapter 12 Solutions

To prove: The probability that X is an even integer (including 0) is e−λcoshλ, where X

is a Poisson random variable with parameter λ.