{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Week10Notes

# Week10Notes - Stat 311 Introduction to Mathematical...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}