# 542 I MS Exam Spring 2002 Page 1 of 1

```542 I
MS Exam Spring 2002
Page 1 of 1
Suppose that \ &micro; Bernoulli ˆ &quot;# ‰, Y &micro; Uniform a!&szlig; &quot;b and ^ &micro; Normal a!&szlig; &quot;b are independent random
variables. Define
[ œ \Y  a&quot;  \ 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 &Ograve;\ œ ! and [ Ÿ A&Oacute;
and
T &Ograve;\ œ &quot; and [ Ÿ A&Oacute;
(be sure to cover all cases A Ÿ !&szlig; !  A Ÿ &quot; and A  &quot;).
b) Find expressions for the cdf of [
J[ &ETH;A&Ntilde; œ T &Ograve;[ Ÿ A&Oacute;
and a pdf for [
0[ &ETH;A&Ntilde;
(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
&quot;9&ETH;[ &Ntilde;
if [  ! or if [  &quot;
&THORN;
if !  [  &quot;
The random variable
\2&ETH;[ &Ntilde;
can be written as a function of \ and Y . Do this. Then use your expression to write an integral that gives
E\2&ETH;[ &Ntilde;
(the numerical value of this is &THORN;\$(\$\$ but don't try to evaluate it here).
f) If only [ is observable, 2&ETH;[ &Ntilde; is a sensible &quot;predictor&quot; of \ . For one thing, it has the same mean as \ ,
namely &quot;# (you may assume this without proof). Compare the predictors of \ ,
s &quot; œ 2&ETH;[ &Ntilde;
\
and
s# œ &quot;
\
#
s &quot; ‰ and Eˆ\  \
s # ‰ ).
on the basis of their mean squared differences from \ (the values Eˆ\  \
#
#
```