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Unformatted text preview: STAT511 — Summer 2009 Lecture Notes 1 Chapter 5 July 7, 2009 Joint Probability Distributions and Random Samples 5.1 Jointly Distributed Random Variables Chapter Overview • Jointly distributed rv – Joint mass function, marginal mass function for discrete rv – Joint density function, marginal density function for continuous rv – Independent random variables • Expectation, covariance and correlation between two rvs – Expectation – Covariance – Correlation – Interpretations • Statistics and their distributions • Distribution of the sample mean • Distribution of a linear combination Joint Mass Function of Two Discrete RVs Definition 1. Let X and Y be two discrete rvs defined on the sample space S of a random experiment. The joint probability mass function p ( x,y ) is defined for each pair of numbers ( x,y ) by: p ( x,y ) = P ( X = x and Y = y ) Let A be the set consisting of pairs of ( x,y ) values, then the probability P [( X,Y ) ∈ A ] is obtained by summing the joint pmf pairs in A : P [( X,Y ) ∈ A ] = X ( x,y ) X ∈ A p ( x,y ) Example of Joint PMF Example 5.1.1 Exercise 5.3: A market has two check out lines. Let X be the number of customers at the express checkout line at a particular time of day. Let Y denote the number of customers in the superexpress line at the same time. The joint pmf of ( X,Y ) is given below: x = ,y = 1 2 3 0.08 0.07 0.04 0.00 1 0.06 0.15 0.05 0.04 2 0.05 0.04 0.10 0.06 3 0.00 0.03 0.04 0.07 4 0.00 0.01 0.05 0.06 What is P ( X = 1 ,Y = 0)? What is P ( X = 1 ,Y > 2)? What is P ( X = Y )? Purdue University Chapter5˙print.tex; Last Modified: July 7, 2009 (W. Sharabati) STAT511 — Summer 2009 Lecture Notes 2 Marginal Probability Mass Function Definition 2. The marginal probability mass functions of X and Y , denoted p X ( x ) and p Y ( y ), respectively, are given by p X ( x ) = P ( X = x ) = X y p ( x,y ) p Y ( y ) = P ( Y = y ) = X x p ( x,y ) Example of Marginal Probability Mass Function Example 5.1.1 Now let’s find the marginal mass function. x = ,y = 1 2 3 p ( x ) 0.08 0.07 0.04 0.00 0.19 1 0.06 0.15 0.05 0.04 0.30 2 0.05 0.04 0.10 0.06 0.25 3 0.00 0.03 0.04 0.07 0.14 4 0.00 0.01 0.05 0.06 0.12 p ( y ) 0.19 0.30 0.28 0.23 1.00 What is P ( X = 3)? What is P ( Y = 2)? Joint Probability Density Function of Two Continuous RVs Definition 3. Let X and Y be continuous rv’s. Then f ( x,y ) is the joint probability density function for X and Y if for any twodimensional set A : P [( X,Y ) ∈ A ] = Z A Z f ( x,y ) dxdy In particular, if A is the twodimensional rectangle { ( x,y ) : a ≤ x ≤ b, c ≤ x ≤ d } , P [( X,Y ) ∈ A ] = Z b a Z d c f ( x,y ) dydx Joint Probability Density Function of Two Continuous RVs P [( X,Y ) ∈ A ] = Volume under density surface above A Purdue University Chapter5˙print.tex; Last Modified: July 7, 2009 (W. Sharabati) STAT511 — Summer 2009 Lecture Notes 3 Marginal Probability Density Function Definition 4. The marginal probability density function of X and...
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This note was uploaded on 03/30/2011 for the course STAT 511 taught by Professor Bud during the Spring '08 term at Purdue.
 Spring '08
 BUD
 Probability

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