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notes-09-15 - M4056 Lecture Notes. September 15, 2010 Some...

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M4056 Lecture Notes. September 15, 2010 Some important distributions Our goal is to investigate the properties of the density function of S 2 n that we discovered last time. This is the so-called chi squared density with n - 1 degrees of freedom. It is best understood in the context of a class of probability densities called the gamma densities . To deFne them, we need to introduce (or renew our acquaintance with) an interesting function. (See page 99.) The gamma function , Γ, is deFned Γ( α ) = i 0 t α - 1 e - t dt. Homework 1. a) Show that Γ(1) = 1. b) Using integration by parts, show Γ( α + 1) = α Γ( α ). c) Using a) and b), show that Γ( n ) = ( n - 1)!. The gamma density with parameters α > 0 and β = 1 is deFned to be: g ( t | α, 1) = 1 Γ( α ) t α - 1 e - t , 0 < t < . Homework 2. a) Verify that g ( t | α, 1) is a probability density. b) Suppose T is has a gamma distribution with parameters α and 1. Show that X = βT has the following distribution. (See page 99, (3.3.6).)

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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

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notes-09-15 - M4056 Lecture Notes. September 15, 2010 Some...

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