Week_7_Slides - Week 7 Discrete Multivariate Distributions...

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Week 7 Discrete Multivariate Distributions Independence Conditional Expectations Law of Iterated Expectation
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Discrete Multivariate Distributions Example 1 A die is rolled. Let X = no. of 6’s and Y = no. of even numbers. Find the joint probability distribution of X and Y . P ( X = 1, Y = 1) = P (6) = 1/6 P ( X = 0, Y = 1) = P (2,4) = 2/6 = 1/3 P ( X = 0, Y = 0) = P (1,3,5) = 3/6 = 1/2. We say that X and Y have a joint probability distribution . The joint pdf of X and Y is 1/ 2, 0 ( , ) ( , ) 1/ 3, 0, 1 1/ 6, 1 x y p x y P X x Y y x y x y
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Joint PMF (or PDF) Graphical Representation Table of p ( x , y ) : y 0 1 x 0 1/2 1/3 1 1/6 Graph (two-dimensional top view):
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Properties of Joint PMF’s Two properties of discrete joint pdfs: 1. ( , ) 0 p x y for all x and y 2. , ( , ) 1 x y p x y . [Equivalently, ( , ) 1 x y p x y   .] [Eg, 1/2 + 1/3 + 1/6 = 1.]
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Joint CDF’s The joint cdf of X and Y is ( , ) ( , ) F x y P X x Y y . In our example observe that: F (0,0) = P ( X 0, Y 0) = p (0,0) = 1/2 F (0,0.5) = P ( X 0, Y 0.5) = p (0,0) = 1/2 F (0,1) = P ( X 0, Y 1) = p (0,0) + p(0,1) = 1/2 + 1/3 = 5/6 F (0.5,1.5) = P ( X 0.5, Y 1.5) = p (0,0) + p(0,1) = 1/2 + 1/3 = 5/6, etc. [
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Joint CDF’s We find that X and Y have joint cdf 0, 0 or 0 1/ 2, 0,0 1 ( , ) 5/ 6, 0 1, 1 1, , 1 x y x y F x y x y x y Graph (top view): [ 3-d graph: F ( x , y )
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Marginal PMF’s ( ) ( , ) y p x p x y . This pdf defines the marginal probability distribution of X . We may also write
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Week_7_Slides - Week 7 Discrete Multivariate Distributions...

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