jan2p - EXAM P QUESTIONS OF THE WEEK S. Broverman, 2006...

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EXAM P QUESTIONS OF THE WEEK S. Broverman, 2006 Week of January 2/06 The continuous random variable is defined on the interval . \ Ð!ß "Ñ For each number in , you are given the conditional expectation + Ð!ß "Ñ IÒ\l\ Ÿ +Ó œ $+ % . If , find . !+" IÒ\l\ +Ó Hint: You may need to use the following version of Leibniz's rule: .. .D !! DD '' 2ÐBß DÑ .B œ 2ÐDß DÑ  Ò 2ÐBß DÑÓ .B . The solution can be found below.
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Week of January 2/06 - Solution The conditional pdf of given is , and the conditional expectation is \\ Ÿ + 0ÐBÑ JÐ+Ñ IÒ\l\ Ÿ +Ó œ B † .B œ ' ! + % $+ . .$ . . .+ % .+ .+ !! ++ 0ÐBÑ +0Ð+Ñ 0ÐBÑ IÒ\ l\ Ÿ +Ó œ œ ÒB † Ó.B '' . Since , we get . .+ ÐJÐ+ÑÑ 0ÐBÑ B0ÐBÑ0Ð+Ñ ÒB † Ó œ  # ' ! + . .+ ÐJÐ+ÑÑ 0ÐBÑ B0ÐBÑ0Ð+Ñ 0Ð+Ñ 0ÐBÑ ÒB † Ó.B œ  .B œ  .B # . œ  †IÒ\l\ Ÿ+Ó œ  0Ð+Ñ % $+ Therefore, , so that . $$ + $ % % %JÐ+Ñ + +0Ð+Ñ 0Ð+Ñ +0Ð+Ñ 0Ð+Ñ œ œ œ Noting that , we have , and .+ + .. $ œ 68JÐ+Ñ 68JÐ+Ñœ 68 JÐ+Ñ œ $ 68 +  - JÐ+Ñ œ + † / - , and (where is a constant). $-
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jan2p - EXAM P QUESTIONS OF THE WEEK S. Broverman, 2006...

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