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Practice Problems for the Final
Problem 1
The three doors on the Monty Hall show open to show a goat; another goat; and a
new car. If a goat cost $1,000 and a car cost $10,000, and if each contestant
chooses a door at random (probability 1/3 each) and sticks with it (no switching),
(a) What is the expected value of one contestant's prize,
X
?
(b) What is the variance?
(c) Let
X
n
be the prize won by the
nth
contestant and let
S
100
=
X
1
+
…
+
X
100
be the
sum. In 100 shows, how much money should Monty Hall's producers expect to
spend on prizes? What would be the variance?
(d) Approximately what is the probability that the prize total will exceed $350,000?
Problem 2
The random variables
X; Y
have the joint probability density:
0
2
( , )
0
c for
y
x
f x y
otherwise
<
< <
=
(a) Determine the constant
c
.
(b) Find the marginal densities
f
X
(
x
) and
f
Y
(
y
).
(c) Find the conditional densities
f
(
xy
) and
f
(
yx
).
(d) Find
P
(
X >
3/2).
(e) Find
P
(
X  Y >
1).
(f) Find
P
(
X >
3/2
Y
= 1)
Problem 3
Suppose that the joint density of random variables
X
and
Y
is:
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2
0
1,0
1
( , )
3
0
x
y
for
x
y
f x y
otherwise
+
< <
<
<
=
Find
( , )
Cov X Y
.
Problem 4
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 Fall '08
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