Answered: .) Prove that the moment generating…

.) Prove that the moment generating function for a Poisson distribution with parameter λ is eλ(e^(t)−1). Hint: Recall the power series expansion for ex. Also note that etkλk = (λet)k.

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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Question

1.) Prove that the moment generating function for a Poisson distribution with parameter λ is e^{λ(e^(t)−1)}. Hint: Recall the power series expansion for e^x. Also note that e^tkλ^k = (λe^t)^k.

Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.

Expert Solution

Step 1

Let, X~ poisson( $λ$ )

The probability mass function is,

$p (X) = \frac{e^{- λ} * λ^{x}}{x!}; x = 0, 1, 2, . . . . . . = 0; o t h e r w i s e W h e r e, λ > 0$

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