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SIEO 3600 (IEOR Majors)
Assignment #7
Introduction to Probability and Statistics
March 9, 2010
Assignment #7
– due March 9th, 2010
(
Expectation, variance and covariance of random variables
)
1. Ten balls are randomly chosen from an urn containing 17 white and 23 black balls. Let
X
denote the number of white balls chosen.
(a) If everytime we put the chosen ball back to the urn (
with replacement
), compute
E
[
X
]
by deﬁning appropriate indicator variables
X
i
,
i
= 1
,...,
10 so that
X
=
10
X
i
=1
X
i
.
(b) If everytime we don’t put the ball back (
without replacement
), compute
E
[
X
] by
deﬁning appropriate indicator variables
Y
i
,
i
= 1
,...,
17 so that
X
=
17
X
i
=1
Y
i
.
Solution:
(a) For
i
= 1
,...,
10, let
X
i
=
±
1
if the
i
th pick is a white ball,
0
otherwise.
With replacement, everything we have the same number of white and black balls.
Hence,
E
(
X
i
) =
P
(
X
i
= 1) = 17
/
40. Linearity of expectation implies that
E
(
X
) =
∑
10
i
=1
E
(
X
i
) = 10
·
E
(
X
i
) = 17
/
4.
(b) With all 17 white balls labeled with
j
= 1
,...,
10, let
Y
j
=
±
1
if white ball number
j
is picked,
0
otherwise.
Without replacement, this is equivalent to picking 10 balls out of 40 at one time. Note
that
E
(
Y
j
) =
P
(
Y
j
= 1) =
P
(white ball number
j
is picked)
=
total number of ways to pick 10 balls out of 40 with white ball number
j
included
total number of ways to pick 10 balls out of 40
=
(
39
9
)
(
40
10
)
=
1
4
.
Hence, linearity of expectation implies that
E
(
X
) =
∑
10
j
=1
E
(
Y
j
) = 17
·
E
(
Y
j
) = 17
/
4.
Although the polices in (a) and (b) are diﬀerent (with or with replacement), the answers
are the same. (Unbelievable!)
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SIEO 3600, Assignment #7
2. Let
X
be a continuous random variable with CDF
F
and PDF
f
, then deﬁne its
median
as
the value
m
such that
F
(
m
) = 1
/
2. (Think about the sample median of a data set). The
median
, like the mean, is important in predicting the value of a random variable. It can
be shown that the mean
E
(
x
) is the best predictor from the point of view of minimizing the
expected value of the square of the error. Prove that the
median
is the best predictor if one
wants to
minimize the expected value of the absolution error
. That is,
E
[

X

c

] is minimized
when
c
is the median of
X
.
Hint:
Write
E
[

X

c

] =
Z
∞
∞

x

c

f
(
x
)
dx
=
Z
c
∞

x

c

f
(
x
)
dx
+
Z
∞
c

x

c

f
(
x
)
dx
=
Z
c
∞
(
c

x
)
f
(
x
)
dx
+
Z
∞
c
(
x

c
)
f
(
x
)
dx
=
cF
(
c
)

Z
c
∞
xf
(
x
)
dx
+
Z
∞
c
xf
(
x
)
dx

c
[1

F
(
c
)]
,
and then choose the best
c
to minimize
E
[

X

c

].
Solutions
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