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Week10Notes - Stat 311 Introduction to Mathematical...

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Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA November 3-5, 2009 Functions of a Random Variable Theorem 5.1 Let X be a continuous random variable, and suppose that φ ( x ) is a strictly increasing function on the range of X . Define Y = φ ( X ). Suppose that X and Y have cumulative distribution functions F X and F Y respectively. Then these functions are related by F Y ( y ) = F X ( φ - 1 ( y )) . If φ ( x ) is strictly decreasing on the range of X , then F Y ( y ) = 1 - F X ( φ - 1 ( y )) . Corollary 5.1 Let X be a continuous random variable, and suppose that φ ( x ) is a strictly increasing function on the range of X . Define Y = φ ( X ). Suppose that the density functions of X and Y are f X and f Y , respectively. Then these functions are related by f Y ( y ) = f X ( φ - 1 ( y )) d dy φ - 1 ( y ) . If φ ( x ) is strictly decreasing on the range of X , then f Y ( y ) = - f X ( φ - 1 ( y )) d dy φ - 1 ( y ) . Corollary 5.2 If F ( y ) is a given cumulative distribution function that is strictly increasing when 0 < F ( y ) < 1 and if U is a random variable with uniform distribution on [0 , 1], then Y = F - 1 ( U ) has the cumulative distribution F ( y ). Simulation 0
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This corollary tells that to simulate a random variable with a given cumulative distri- bution F we need to simulate a uniform random variable first, then convert the value into the desired scale using the inverse distribution function. Example A number U is chosen at random in the interval [0 , 1]. Find the distribution functions and density functions for the following transformed random variables. 1. R = U 2 . 2. S = U (1 - U ). 3. T = U/ (1 - U ). Normal density: The bell shape curve -4 -2 2 4 0.1 0.2 0.3 0.4 σ=1 σ=2 Figure 1: Normal density function f X ( x ) = 1 2 πσ e - ( x - μ ) 2 / 2 σ 2 .
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