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542 I MS Exam Spring 2002 Page 1 of 1 Suppose that \ µ Bernoulli ˆ "# ‰, Y µ Uniform a!ß "b and ^ µ Normal a!ß "b are independent random variables. Define [ œ \Y a" \ b^ (so that [ is either Y or ^ , depending upon the value of \ ). In your answers to what follows, use F for the standard normal cdf and 9 for the standard normal pdf. a) Find expressions for T Ò\ œ ! and [ Ÿ AÓ and T Ò\ œ " and [ Ÿ AÓ (be sure to cover all cases A Ÿ !ß ! A Ÿ " and A "). b) Find expressions for the cdf of [ J[ ÐAÑ œ T Ò[ Ÿ AÓ and a pdf for [ 0[ ÐAÑ (again be sure to cover all cases for A). (Hint: Part a) is relevant here.) c) Use the notion of conditioning to find numerical values for the mean and variance of [ , E[ and Var[ . d) Evaluate the correlation between [ and ^ . e) Consider the function of [ 2a[ b œ 0 1 "9Ð[ Ñ if [ ! or if [ " Þ if ! [ " The random variable \2Ð[ Ñ can be written as a function of \ and Y . Do this. Then use your expression to write an integral that gives E\2Ð[ Ñ (the numerical value of this is Þ$($$ but don't try to evaluate it here). f) If only [ is observable, 2Ð[ Ñ is a sensible "predictor" of \ . For one thing, it has the same mean as \ , namely "# (you may assume this without proof). Compare the predictors of \ , s " œ 2Ð[ Ñ \ and s# œ " \ # s " ‰ and Eˆ\ \ s # ‰ ). on the basis of their mean squared differences from \ (the values Eˆ\ \ # #