Elementary Distribution Theory: Transforming a Bivariate Distribution

Tutorial 9, STAT1301 Fall 2010, 23NOV2010, MB103@HKUBy Joseph Dong

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A Guide

•This topic is mainly technical, you should focus on the handout problems more than the slides.▫Try to solve them multiple times until you

experience the Eureka.•Try more exercises elsewhere such as in

lecture notes and slides, assignments, and extra exercises.

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Statement of The Prime Problem

• How to find the joint distribution of a random vector that is dependent in a known way to another random vector whose joint distribution is already known.

• E.g.: ▫ We want to find the joint distribution of the random

vector . We also know that depends on another random vector in the following functional way:

▫ In addition, the joint distribution of has already been found as bivariate uniform distribution in the unit square. Are we able to find the support of and on it the joint density now?

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Two MethodsMethod 1: Using Calculus Method 2: Using Probability

Based on

• Require 1-1 transformation

• Require non-singular Jacobian

• Jacobian of over refers to the matrix

Based on

• Generally applicable but more complicated.

• E.g.:

This integration will give you a function of only. The result of the integration is the cdf of the random variable evaluated at , differentiate it wrt will get you the pdf. If your ultimate goal is the joint pdf of , then this has found you the marginal pdf of already.

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Method 1: Using Calculus• Step 1:

▫Find the support of from the support of . Divide the support if necessary.

• Step 2:▫ If the transformation is obvious not 1-1, then

abandon it and switch to Method 2.▫Derive the Jacobian matrix and then calculate its

determinant. If the determinant is zero, then abandon it and switch to Method2.

• Step 3: ▫Apply the formula and summarize the final pdf

found.

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Method 2: Using Probability• Step 1:

▫ Find the support of from the support of . Divide the support if necessary.

• Step 2:▫ Draw the region of on the support of .

• Step 3:▫ Find the marginal cdf of by integrating using the joint

density of on .• Step 4:

▫ Differentiate the marginal cdf of to get its marginal pdf.• Step 5, 6, 7:

▫ Repeat steps 2,3,4 for .

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Key Technical Skills

•Solving systems of inequalities•Graphing systems of inequalities•Integration•Differentiation

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Handouts

•Review part:▫The 4 arithmetic transformations of two

random variables.•Problems