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Let X~ Gamma(a, B) be a Gamma random variable. That is, fx (x) = O-le-/ 0, %3D I< 0. Consider the reciprocal variable, Y . %3D (a) Consider the cumulative distribution function (cdf) ofY, FY(y). Write Fy (y) in terms of the cdf of X, Fx. (b) Verify that the cdf you derived above is a valid cdf.

Let X~ Gamma(a, B) be a Gamma random variable. That is, fx (x) = O-le-/ 0, %3D I< 0. Consider the reciprocal variable, Y . %3D (a) Consider the cumulative distribution function (cdf) ofY, FY(y). Write Fy (y) in terms of the cdf of X, Fx. (b) Verify that the cdf you derived above is a valid cdf.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 19E
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please send solution for part a and b
Question 3 Let X~ Gamma(a, B) be a Gamma random variable. That is, >0; fx (x) = { rOp-e-=/8 %3D I< 0. Consider the reciprocal variable, Y = x. (a) Consider the cumulative distribution function (cdf) of Y, Fy(y). Write Fy (y) in terms of the cdf of X, Fx. (b) Verify that the cdf you derived above is a valid cdf. (c) Consider the probability density function (pdf) of Y, fy(y). Show that the pdf of Y is, fy(y) = y > 0; y<0. (d) Verify that the pdf above is a valid pdf. (e) Find the moments of Y directly by evaluating, !! where k is a positive integer. (f) Show that the moment generating function (mgf) of Y does not exist. That is, My (t) = E(e") = | c"f(v) dy, %3D does not exist.
Transcribed Image Text:Question 3 Let X~ Gamma(a, B) be a Gamma random variable. That is, >0; fx (x) = { rOp-e-=/8 %3D I< 0. Consider the reciprocal variable, Y = x. (a) Consider the cumulative distribution function (cdf) of Y, Fy(y). Write Fy (y) in terms of the cdf of X, Fx. (b) Verify that the cdf you derived above is a valid cdf. (c) Consider the probability density function (pdf) of Y, fy(y). Show that the pdf of Y is, fy(y) = y > 0; y<0. (d) Verify that the pdf above is a valid pdf. (e) Find the moments of Y directly by evaluating, !! where k is a positive integer. (f) Show that the moment generating function (mgf) of Y does not exist. That is, My (t) = E(e") = | c"f(v) dy, %3D does not exist.
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