# Probability Distribution Function (PDF) for a Discrete Random Variable

### Learning Outcomes

• Recognize and understand discrete probability distribution functions, in general

The idea of a random variable can be confusing. In this video we help you learn what a random variable is, and the difference between discrete and continuous random variables.

A discrete probability distribution function has two characteristics:

1. Each probability is between zero and one, inclusive.
2. The sum of the probabilities is one.

### Example

A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let
$X=$
the number of times per week a newborn baby's crying wakes its mother after midnight. For this example,
$x = 0, 1, 2, 3, 4, 5$
.

$P(x) =$
probability that
$X$
takes on a value
$x$
.

$x$
$P(x)$
$0$
$P(x = 0)$
$=$
$(\frac{2}{50})$
$1$
$P(x = 1)$
$=$

$(\frac{11}{50})$
$2$
$P(x = 2) =$

$(\frac{23}{50})$
$3$
$P(x = 3) =$

$(\frac{9}{50)}$
$4$
$P(x = 4) =$

$(\frac{4}{50})$
$5$
$P(x = 5) =$

$(\frac{1}{50})$

$X$
takes on the values
$0, 1, 2, 3, 4, 5.$
This is a discrete PDF because:

1. Each
$P(x)$
is between zero and one, inclusive.
2. The sum of the probabilities is one, that is,

$(\frac{2}{50})+(\frac{11}{50})+(\frac{23}{50})+(\frac{9}{50})+(\frac{4}{50})+(\frac{1}{50})=1$

### Try it

Suppose Nancy has classes three days a week. She attends classes three days a week
$80$
%
of the time, two days
$15$
%
of the time, one day
$4$
%
of the time, and no days
$1$
of the time. Suppose one week is randomly selected.

a. Let
$X$
= the number of days Nancy ____________________.
b.
$X$
takes on what values?
c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in Example 1. The table should have two columns labeled
$x$
and
$P(x)$
. What does the
$P(x)$
column sum to?

### Example

Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice. What is X and what values does it take on?

Solution:

$X$
is the number of days Jeremiah attends basketball practice per week.
$X$
takes on the values
$0, 1,$
and
$2.$

## Concept Review

The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows:

1. Each probability is between zero and one, inclusive (inclusive means to include zero and one).
2. The sum of the probabilities is one.

### Try it

Solution:

a. Let
$X$
= the number of days Nancy attends class per week.

b.
$0, 1, 2,$
and
$3$

c.

$x$
$P(x)$
$0$
$0.01$
$1$
$0.04$
$2$
$0.15$
$3$
$0.80$