542 I
MS Exam Spring 2002
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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\  \
#
#