SIEO 3600 (IEOR Majors)
Assignment #6
Introduction to Probability and Statistics
February 24, 2010
Assignment #6
– due March 2nd, 2010
(
Expectation and variance of random variables
)
1. (continued with Problem 2 in HW5)
X
and
Y
are two continuous random variables with joint
probability density function
f
X,Y
(
x, y
) =
10
xy
2
,
0
< x < y <
1;
0
,
otherwise
.
(a) Compute
E
[
X
] and
E
[
Y
]. (
Hint
: You have calculated the marginal PDF of
X
and
Y
in
HW5)
(b) Compute
E
[
X
2
] and
E
[
Y
2
].
(c) Use (a) and (b) to compute var(
X
) and var(
Y
).
2. Suppose that
X
and
Y
are independent continuous random variables. Show that
(a)
P
(
X
+
Y
≤
a
) =
R
∞
∞
F
X
(
a

y
)
f
Y
(
y
)
dy
(b)
P
(
X
≤
Y
) =
R
∞
∞
F
X
(
y
)
f
Y
(
y
)
dy
where
f
Y
is the density function of
Y
, and
F
X
is the distribution function of
X
.
3. Each night different meteorologists give us the “probability” that it will rain the next day. To
judge how well these people predict, we will score each of them as follows: If a meteorologist
says that it will rain with probability
p
, then he or she will receive a score of
1

(1

p
)
2
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 Spring '10
 YUNANLIU
 Probability theory, probability density function, IEOR Majors

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