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Midterm2 - Mathematics Department UCLA T Richthammer spring...

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Mathematics Department, UCLA T. Richthammer spring 09, midterm 2 May 18, 2009 Midterm 2: Math 170B Probability, Sec. 1 1. Let X, Y be continuous RVs. (a) What is the conceptual difference between E ( X | Y = y ) and E ( X | Y )? (b) Give the definition of E ( X | Y ) and show that E ( E ( X | Y )) = E ( X ). Answer: (a) E ( X | Y = y ) is a number for every choice of y , whereas E ( X | Y ) is a function of Y and thus a RV (which has certain values with certain probabilities). (b) We have E ( X | Y ) = h ( Y ), where h ( y ) = E ( X | Y = y ) = R xf X ( x | Y = y ) dx = R x f X,Y ( x,y ) f Y ( y ) dx . So E ( E ( X | Y )) = E ( h ( Y )) = R h ( y ) f Y ( y ) dy = R ( R x f X,Y ( x,y ) f Y ( y ) dx ) f Y ( y ) dy = R xf X,Y ( x, y ) dxdy = E ( X ). 2. You order an item at time 0. It takes a random time T until the item is delivered. At a random time X between 0 and T you get an email confirming your order. Suppose that T is a RV with E ( T ) = 6 and V ( T ) = 3. Calculate the expectation and variance of the time X you get the confirmation email in the two cases (a) and (b). (a) Assume that
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Midterm2 - Mathematics Department UCLA T Richthammer spring...

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