173Lecture4 - Lecture 4 Mariana Olvera-Cravioto UC Berkeley...

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Lecture 4 Mariana Olvera-Cravioto UC Berkeley [email protected] January 26th, 2017 IND ENG 173, Introduction to Stochastic Processes Lecture 4 1/12
More on conditional expectation and variance I The conditional expectation and the conditional variance of X given Y = y are functions of y : g ( y ) = E [ X | Y = y ] and h ( y ) = Var( X | Y = y ) I E [ X | Y ] , g ( Y ) and Var( X | Y ) , h ( Y ) are random variables! I E [ X | Y ] is called the conditional expectation of X given Y , and it satisfies: E [ E [ X | Y ]] = E [ g ( Y )] = X y g ( y ) p Y ( y ) = X y E [ X | Y = y ] p Y ( y ) = E [ X ] I Var( X | Y ) is called the conditional variance of X given Y . However, E [Var( X | Y )] 6 = Var( X )!!! IND ENG 173, Introduction to Stochastic Processes Lecture 4 2/12
Total variance formula I It is not hard to check that Var( X | Y ) = E [ X 2 | Y ] - ( E [ X | Y ]) 2 I Taking expectations on both sides gives E [Var( X | Y )] = E [ E [ X 2 | Y ]] - E [( E [ X | Y ]) 2 ] = E [ X 2 ] - E [( E [ X | Y ]) 2 ] I Similarly, Var( E [ X | Y ]) = E [( E [ X | Y ]) 2 ] - ( E [ E [ X | Y ]]) 2 = E [( E [ X | Y ]) 2 ] - ( E [ X ]) 2 I Summing the last two equations gives E [Var( X | Y )] + Var( E [ X | Y ]) = E [ X 2 ] - ( E [ X ]) 2 = Var( X ) IND ENG 173, Introduction to Stochastic Processes Lecture 4 3/12
Random sums I On a given day a supermarket has N customers. Each customer spends a random amount of money X i . I Let S be the total amount of sales on a given day. S = N X i =1 X i I Assume that the X i ’s are iid and independent of N . I Compute E [ S ] and Var( S ) . IND ENG 173, Introduction to Stochastic Processes Lecture 4 4/12
Random sums I On a given day a supermarket has N customers. Each customer spends a random amount of money X i . I Let S be the total amount of sales on a given day. S = N X i =1 X i I Assume that the X i ’s are iid and independent of N . I Compute E [ S ] and

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