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Homework Assignment 1
1. Each of the following nonnegative random variables
X
has a continuous distribution. You
are given either the density function
f
(
x
) or the cumulative distribution function (cdf)
F
(
x
) =
P
(
X
≤
x
). From whichever you are given, derive the other so you have both.
Then calculate the expected value
E
(
X
) in two ways: Directly via
E
(
X
) =
R
xf
(
x
)
dx
and by integrating the tail
E
(
X
) =
R
P
(
X > x
)
dx
.
(a) (exponential distribution) With
λ >
0 a ﬁxed constant
X
has density function
f
(
x
) =
±
λe

λx
;
if
x
≥
0
0
,
if
x <
0.
(b) (uniform distribution on (
a,b
); 0
≤
a < b
)
X
has density function
f
(
x
) =
±
1
b

a
;
if
x
∈
(
a,b
)
0
,
otherwise.
(c) (Pareto distribution)
X
has cdf
F
(
x
) =
±
0;
if
x <
1
1

1
x
3
,
if
x
≥
1 .
2. (Continued) Now repeat the above for calculating the second moment
E
(
X
2
) in two ways:
Directly via
E
(
X
2
) =
R
x
2
f
(
x
)
dx
and by use of
E
(
X
2
) =
Z
∞
0
2
xP
(
X > x
)
dx.
(1)
3. (Continued) Using your answers from above, give the variance
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 Fall '08
 Whitt
 Operations Research

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