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(a) Suppose that X1,..., X, are independent Laplace random variables, and let Y, = X1 + ……+ Xn. Find the mean, variance, and mgf of Yn. (b) Let Yn – HY. be the standardized version of Yn. Find the mgf of Zn.

(a) Suppose that X1,..., X, are independent Laplace random variables, and let Y, = X1 + ……+ Xn. Find the mean, variance, and mgf of Yn. (b) Let Yn – HY. be the standardized version of Yn. Find the mgf of Zn.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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3. A random variable X has the Laplace distribution if it follows the pdf S(2) = H, On the midterm exam you verified that the mgf of a Laplace distributed X is 1 -00 <x < 0. 1 Mx(t) 1- t2 and so its mean is 0 and its variance is 2. (a) Suppose that X1,...,X, are independent Laplace random variables, and let Y, = X1+...+ Xn. Find the mean, variance, and mgf of Y,. (b) Let Yn – HYn Zn be the standardized version of Yn. Find the mgf of Zn. (c) Show that in the limit as n → 00, the mgfs of Zn converge to the mgf of a standard normal random variable. (This is a version of the Central Limit Theorem.)
Transcribed Image Text:3. A random variable X has the Laplace distribution if it follows the pdf S(2) = H, On the midterm exam you verified that the mgf of a Laplace distributed X is 1 -00 <x < 0. 1 Mx(t) 1- t2 and so its mean is 0 and its variance is 2. (a) Suppose that X1,...,X, are independent Laplace random variables, and let Y, = X1+...+ Xn. Find the mean, variance, and mgf of Y,. (b) Let Yn – HYn Zn be the standardized version of Yn. Find the mgf of Zn. (c) Show that in the limit as n → 00, the mgfs of Zn converge to the mgf of a standard normal random variable. (This is a version of the Central Limit Theorem.)
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