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!)