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Show that the random variable Y = X² has gamma distribution with shape parameter a = 1/2 and scale parameter 3 = 1/2 (see the handout on stat distributions for the notation). Note: This particular gamma

Show that the random variable Y = X² has gamma distribution with shape parameter a = 1/2 and scale parameter 3 = 1/2 (see the handout on stat distributions for the notation). Note: This particular gamma

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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2. (Stat 467 only) Suppose that X has standard normal distribution. Show that the random variable Y = X² has gamma distribution with shape parameter a = 1/2 and scale parameter 3 = 1/2 (see the handout on stat distributions for the notation). Note: This particular gamma distribution is known as chi-square distribution with 1 degree of free- dom. A more general chi-square distribution with k degrees of freedom, where k is a positive integer, is gamma distribution with shape param- eter a = k/2 and scale parameter 3 = 1/2. Hint: First, derive the cu- mulative distribution function (CDF) of Y, Fy (y) (where you carefully distinguish the cases y < 0 and y ≥ 0), and then take the derivative of Fy(y) to obtain the PDF. Also, use the fact that I'(1/2) = √√√.
Transcribed Image Text:2. (Stat 467 only) Suppose that X has standard normal distribution. Show that the random variable Y = X² has gamma distribution with shape parameter a = 1/2 and scale parameter 3 = 1/2 (see the handout on stat distributions for the notation). Note: This particular gamma distribution is known as chi-square distribution with 1 degree of free- dom. A more general chi-square distribution with k degrees of freedom, where k is a positive integer, is gamma distribution with shape param- eter a = k/2 and scale parameter 3 = 1/2. Hint: First, derive the cu- mulative distribution function (CDF) of Y, Fy (y) (where you carefully distinguish the cases y < 0 and y ≥ 0), and then take the derivative of Fy(y) to obtain the PDF. Also, use the fact that I'(1/2) = √√√.
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