Chapter5_print

Chapter5_print - STAT511 Summer 2009 Lecture Notes 1...

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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 super-express 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 Chapter5print.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 lets 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 rvs. Then f ( x,y ) is the joint probability density function for X and Y if for any two-dimensional set A : P [( X,Y ) A ] = Z A Z f ( x,y ) dxdy In particular, if A is the two-dimensional 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 Chapter5print.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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Chapter5_print - STAT511 Summer 2009 Lecture Notes 1...

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