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1. Let X₁,..., X~ Poi(2) be independent Poisson random variables with parameter and probability mass function p(x; λ) = -,x= {0,...,00}. λxe-a x! (a) Derive the moment generating function for the Xi's. Useful fact: the power series expansion for the exponential function is =[²². n! e² = [ n=0 (b) Use the moment generating function to show that E(X;) = 2 and Var(Xi) = 2. (c) We know that the sum of n iid Poisson distributed random variable still follows a Poisson distribution, use the moment generating function to find the parameter for Sn = [/1 X₁. (d) If λ = 5 and n = 25, use central limit theory to approximate P(Xn> 4.5) where XnSn denotes the sample mean.

1. Let X₁,..., X~ Poi(2) be independent Poisson random variables with parameter and probability mass function p(x; λ) = -,x= {0,...,00}. λxe-a x! (a) Derive the moment generating function for the Xi's. Useful fact: the power series expansion for the exponential function is =[²². n! e² = [ n=0 (b) Use the moment generating function to show that E(X;) = 2 and Var(Xi) = 2. (c) We know that the sum of n iid Poisson distributed random variable still follows a Poisson distribution, use the moment generating function to find the parameter for Sn = [/1 X₁. (d) If λ = 5 and n = 25, use central limit theory to approximate P(Xn> 4.5) where XnSn denotes the sample mean.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!

1. Let X₁,..., X₂~ Poi(2) be independent Poisson random variables with parameter and probability mass function p(x; 2) = -₂x = {0,..., ∞}. λxe-a x! (a) Derive the moment generating function for the Xi's. Useful fact: the power series expansion for the exponential function is e¹ = = Σ n! n=0 (b) Use the moment generating function to show that E(X;) = 2 and Var(Xi) = 2. (c) We know that the sum of n iid Poisson distributed random variable still follows a Poisson distribution, use the moment generating function to find the parameter for Sn = [²_₁ X₁. (d) If λ = 5 and n = 25, use central limit theory to approximate P(X₂ ≥ 4.5) where X₂ = S₂ denotes the sample mean.
Transcribed Image Text:1. Let X₁,..., X₂~ Poi(2) be independent Poisson random variables with parameter and probability mass function p(x; 2) = -₂x = {0,..., ∞}. λxe-a x! (a) Derive the moment generating function for the Xi's. Useful fact: the power series expansion for the exponential function is e¹ = = Σ n! n=0 (b) Use the moment generating function to show that E(X;) = 2 and Var(Xi) = 2. (c) We know that the sum of n iid Poisson distributed random variable still follows a Poisson distribution, use the moment generating function to find the parameter for Sn = [²_₁ X₁. (d) If λ = 5 and n = 25, use central limit theory to approximate P(X₂ ≥ 4.5) where X₂ = S₂ denotes the sample mean.
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