MA/STAT416: Probability
Lecture Notes
Spring 2011
1
Chapter 4: Discrete Random Variables and Mass Functions
February 28, 2011
4
Expectation
4.4
Mean and Variance
Definition 1.
Let
X
be a discrete random variable taking the values
x
1
, x
2
,
· · ·
, x
n
.
Furthermore, let
p
1
, p
2
,
· · ·
, p
n
be the pmf of
X
. The expected value of
X
is defined by
E
(
X
) =
∑
n
i
=1
x
i
·
p
i
.
Note:
The mean of
X
is
μ
X
=
E
(
X
).
Definition 2.
The variance of
X
is defined by
σ
2
X
=
V ar
(
X
) =
∑
n
i
=1
(
x
i

μ
X
)
2
·
p
i
.
Note:
The variance of
X
is
V ar
(
X
) =
E
(
X
2
)

[
E
(
X
)]
2
=
E
(
X
2
)

μ
2
X
.
Example
3
.
For each of the following random variables, find the mean and variance.
1.
X
= number of heads in 2 tosses of a fair coin.
2.
X
= sum of two fair dice rolls.
3.
X
= larger of two numbers chosen with replacement from 1
,
2
,
· · ·
,
10.
Property:
Let
X
1
,
X
2
be two random variables such that
X
=
X
1
+
X
2
. Then,
E
(
X
) =
E
(
X
1
+
X
2
) =
E
(
X
1
) +
E
(
X
2
).
Example
4
.
3 cookies are distributed completely at random (and independently) to 3
kids. Let
X
be the number of kids who do not receive any cookies. Find the expected
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 Spring '08
 Staff
 Probability, Variance, Probability theory, Purdue University, Prof. Sharabati

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