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a) Find the moment generating function of X~ Bernoulli(p). LE CO

a) Find the moment generating function of X~ Bernoulli(p). LE CO

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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a) Find the moment generating function of X~ Bernoulli(p). b) Find the moment generating function of S~ Binomial(n, p). c) Another useful fact about moment generating functions is the following: Theorem: Suppose (Xn)n>1 is a sequence of random variables with corresponding moment generating functions Mx, and X is a random variable with moment generating function Mx such that for some > 0 we have Mx (1) <∞ for all / € (-6, 6). If for all t, lim Mx, (t) = Mx (1) 72-00 then lim,→∞ Fx₂(x) = Fx(r) for all r where Fx is continuous. That is, if the moment generating functions of X₁ converge to the moment generating function of X, then the distribution of X₁ converges to the distribution of X. Binomial(n,A), then the distribution of S₁, converges to Use this to show that if Sn Poisson(X) as n→∞0.
Transcribed Image Text:a) Find the moment generating function of X~ Bernoulli(p). b) Find the moment generating function of S~ Binomial(n, p). c) Another useful fact about moment generating functions is the following: Theorem: Suppose (Xn)n>1 is a sequence of random variables with corresponding moment generating functions Mx, and X is a random variable with moment generating function Mx such that for some > 0 we have Mx (1) <∞ for all / € (-6, 6). If for all t, lim Mx, (t) = Mx (1) 72-00 then lim,→∞ Fx₂(x) = Fx(r) for all r where Fx is continuous. That is, if the moment generating functions of X₁ converge to the moment generating function of X, then the distribution of X₁ converges to the distribution of X. Binomial(n,A), then the distribution of S₁, converges to Use this to show that if Sn Poisson(X) as n→∞0.
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