STAT 36217, HW 7, due Thursday 10/22/2009, 10:30 AM
PLEAE USE THIS AS THE COVER PAGE
Your Name:
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View Full Document1. Let
X
and
Y
be independent random variables with means
μ
x
and
μ
y
and variances
σ
2
x
and
σ
2
y
. Show that
Var(
XY
) =
σ
2
x
σ
2
y
+
μ
2
y
σ
2
x
+
μ
2
x
σ
2
y
.
2. Let
X
and
Y
be independent normal random variables, each having mean
μ
and
variance
σ
2
. Calculate Cov(
X
+
Y,X

Y
).
3. The joint density of
X
and
Y
is
f
(
x,y
) =
y
2

x
2
8
e

y
,
0
< y <
∞
,

y
≤
x
≤
y.
Show that
E
[
X

y
=
y
] = 0.
4. We start with a stick of length
‘
= 1 . We break it at a point which is chosen according
to a uniform distribution and keep the piece, of length
Y
, that contains the left end
of the stick. We then repeat the same process on the piece that we were left with,
and let
X
be the length of the remaining piece after breaking for the second time.
Find the joint PDF of
Y
and
X
, and the marginal PDF of
X
.
5. A total of 11 people, including you, are invited to a party. The time at which peo
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 Fall '09
 Jin
 Variance, Laplace, Probability theory, two hours, Three hours, joint PDF, conditional distribution

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