1 Discrete Random Variables
We will now review a number of well known discrete random variables. In each case we will begin with an
example. We will then give the deﬁnition, often through the speciﬁcation of the probability mass function,
and will compute the mean and the variance.
1.1 Multinomial
Consider a partition of the sample space
S
into
k
mutually exclusive and collectively exhaustive subsets
A
1
,...,A
k
and let
p
j
=
P
(
A
j
)
≥
0 and
∑
k
j
=1
p
j
= 1. Suppose we repeat the experiment
n
times under
identical conditions. Then we may be interested in the probability that
A
j
occurs
x
j
times for
j
= 1
,...,k
with
∑
k
j
=1
x
j
=
n
.
Example:
Suppose that 10 students are each to independently select one of three sections of a given course.
Suppose that each student has probability 1/3 of selecting Section 1, 1/6 of selecting Section 2, and 1/2
of selecting Section 3. What is the probability that
x
1
= 3 students select Section 1,
x
2
= 2 students
select Section 2, and
x
3
= 5 students select Section 3? Consider the following ordering of the selection:
(1
,
1
,
1
,
2
,
2
,
3
,
3
,
3
,
3
,
3). In this ordering the ﬁrst three students select Section 1, the following two students
select Section 2 and the last ﬁve select Section 3. The probability of this selection is (1
/
3)
3
*
(1
/
6)
2
*
(1
/
2)
5
,
but there are altogether
(
10
3
,
2
,
5
)
= 2520 arrangements that result in the desired number of students in each
section. Thus, the probability is given by
p
(3
,
2
,
5) =
±
10
3
,
2
,
5
¶
(1
/
3)
3
*
(1
/
6)
2
*
(1
/
2)
5
= 0
.
040509259
.
In general, if the sample space is partitioned into