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class01-3-handouts

# class01-3-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 01-3- Method of Moment-Generating Functions Jarad Niemi University of California, Santa Barbara 2 April 2010 Jarad Niemi (UCSB) Method of Moment-Generating Functions 2 April 2010 1 / 16

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Class overview Announcements Announcements Grader: Tony Pourmohamad Homework: Turn homework in either 1) at class or 2) to Tony between 3 and 4pm on Mondays in South Hall 5521 (Rachev room) First homework is due on Monday Order statistic questions have been eliminated Jarad Niemi (UCSB) Method of Moment-Generating Functions 2 April 2010 2 / 16
Class overview Day’s Goals Goals Consider a random variable Y with a known distribution, what is the distribution of U = f ( Y ) for some function f ( · )? Method of Transformations Gamma example Method of Moment-Generating Functions Alternative Distribution Function Method (not in book) Jarad Niemi (UCSB) Method of Moment-Generating Functions 2 April 2010 3 / 16

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Method of transformations Summary Summary of the Transformation Method Let U = h ( Y ), where h ( y ) is either an increasing or decreasing function of y for all y such that f Y ( y ) > 0. 1 Find the inverse function y = h - 1 ( u ). 2 Evaluate d [ h - 1 ( u )] du . 3 Find f U ( u ) = f Y [ h - 1 ( u )] d [ h - 1 ( u )] du . Method of transformations can be extended to functions that are both increasing and decreasing. Jarad Niemi (UCSB) Method of Moment-Generating Functions 2 April 2010 4 / 16
Method of transformations Gamma distribution Gamma distribution See section 4.6. A random variable Y is said to have a gamma distribution with parameters α > 0 and β > 0 if and only if the density function of Y is f ( y ) = ( y α - 1 e - y β α Γ( α ) , 0 y < , 0 , otherwise , where Γ( α ) = Z 0 s α - 1 e - s ds . E ( Y ) = αβ V ( Y ) = αβ 2 If α = v / 2 and β = 2, Y is said to have a χ 2 -distribution with v degrees of freedom .

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class01-3-handouts - PSTAT 120B Probability Statistics...

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