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lecture17 - Stat 302, Introduction to Probability Jiahua...

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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 3 January-April 2011 1 / 17 Example of joint pmf Suppose X and Y have joint pmf given by (after multiplied by 290) y = y = 1 y = 2 y = 3 p X ( x ) x = 3 4 3 x = 1 5 45 45 5 x = 2 90 30 x = 3 30 30 p Y ( y ) 290 What are the marginal pmf’s of X and Y ? Jiahua Chen () Lecture 3 January-April 2011 2 / 17 Marginal pmf’s The answer can be found out by adding up these probabilities row-wise or column-wise: y = y = 1 y = 2 y = 3 p X ( x ) x = 3 4 3 10 x = 1 5 45 45 5 100 x = 2 90 30 120 x = 3 30 30 60 p Y ( y ) 35 168 79 8 290 What are the conditional pmf of X given Y = 2 (or given other values of Y )? Jiahua Chen () Lecture 3 January-April 2011 3 / 17 Conditional pmf’s The answer can be found out dividing each row by its row total: (each row is the value of the conditional probability of X=x given Y = y): y = y = 1 y = 2 y = 3 p X ( x ) P ( X = | Y = y ) 3 / 168 4 / 79 3 / 8 P ( X = 1 | Y = y ) 5 / 35 45 / 168 45 / 79 5 / 8 P ( X = 2 | Y = y ) 90 / 168 30 / 79 P ( X = 3 | Y = y ) 30 / 35 30 / 168 P ( Y = y ) 35 168 79 8 . In particular, P ( X = | Y = 2 ) = 4 / 79, P ( X = 1 | Y = 2 ) = 45 / 79 and P ( X = 2 | Y = 2 ) = 30 / 79. How much is E ( X | Y = 2 ) ? Jiahua Chen () Lecture 3 January-April 2011 4 / 17 Conditional expectation and variance From P ( X = | Y = 2 ) = 4 / 79, P ( X = 1 | Y = 2 ) = 45 / 79 and P ( X = 2 | Y = 2 ) = 30 / 79, we find E ( X | Y = 2 ) = × 4 / 79 + 1 × 45 / 79 + 2 × 30 / 79 = 105 / 79....
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This note was uploaded on 04/07/2011 for the course STAT 302 taught by Professor Dr.chen during the Spring '11 term at UBC.

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lecture17 - Stat 302, Introduction to Probability Jiahua...

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