Stat 542 Exam 2
November 15, 2001
Prof. Vardeman
1. Below is a table specifying the discrete joint distribution of random variables \ and ] .
CÏB
&
%
$
#
"
!
7pts
a) Evaluate T Ò\ € ] ž #Ó.
7pts
b) Evaluate EÒ] l\ œ %Ó.
7 pts
c) Suppose that
!
Þ#&
!
!
Þ!"
Þ!$
Þ!#
"
!
!
Þ!"
Þ!$
Þ!#
Þ!$
\€]
[ œœ
!
#
!
Þ!"
Þ!$
Þ!#
Þ!$
Þ!"
$
Þ!"
Þ!$
Þ!#
Þ!$
Þ!"
!
%
Þ!$
Þ!#
Þ!$
Þ!"
!
!
&
Þ!#
Þ!$
Þ!"
!
!
Þ#&
if both \ and ] are odd
otherwise
Write out a sum that is E[ . (You need not do the arithmetic necessary to simplify this.)
1
2. (Moore) Suppose that in appropriate units, the following is true. The length of a standard bar
of steel is .. A copy of the bar is not perfect, and has length P" so that H" œ P" • . µ Na!ß "b.
A copy of this copy has length P# so that H# œ P# • P" µ Na!ß "b and H# is independent of P" .
A copy of the copy of the copy has length P$ so that H$ œ P$ • P# µ Na!ß "b and H$ is
independent of P" and P# .
10 pts
a) What is the (joint) distribution of P œ aP" ß P# ß P$ bw ?
5 pts
b) Evaluate the correlation between P" and P$ .
5 pts
c) What is the distribution of P$ • ., the error in length of the last bar?
2
3. Suppose that Y µ Uniform a!ß "b and that the conditional distribution of ] |Y œ ? is Na?ß "b.
7pts
a) Evaluate Var] .
7 pts
b) Evaluate CovaY ß ] b.
4. Suppose that \ and ] are jointly "uniform on the unit circle," i.e. jointly continuous with pdf
0 ÐBß CÑ œ œ 1
!
"
if B# € C# • "
otherwise
7 pts
a) For B - Ð • "ß "Ñ, what is the conditional distribution of ] l\ œ B?
5 pts
b) Are \ and ] independent? Explain.
3
20 pts
5. Find a pdf for W œ Y^ where ^ and Y are independent random variables, ^ µ NÐ!ß "Ñ and
Y µ Uniform Ð!ß "Ñ. (Among other possibilities, a transformation will work here.)
4
6. The Poisson a-b moment generating function is
Q a>b œ expa-aexpa>b • "bb
7 pts
(a) If \ µ Poisson a-b, what is the mgf of ^ œ a\ • -bÎÈ- , say Q^ Ð>Ñ?
6 pts
(b) A second order Taylor expansion implies that expÐBÑ œ " € B €
expalBlblBl$
.
'
B#
#
€ VÐBÑ where
lVÐBÑl Ÿ
Apply this to the form you found in part (a) and identify a limit for Q^ Ð>Ñ
as - p _. (Don't approximate the "outside exponential," only its argument.)
5