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Discrete Random Variables: Probability Distribution Function (PDF) for a Discrete Random Variable

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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Summary: This module introduces the Probability Distribution Function (PDF) and its characteristics.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

A discrete probability distribution function has two characteristics:

  • Each probability is between 0 and 1, inclusive.
  • The sum of the probabilities is 1.

P(X)P(X) is the notation used to represent a discrete probability distribution function.

Example 1

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 XX = the number of times a newborn wakes its mother after midnight. For this example, xx = 0, 1, 2, 3, 4, 5.

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

Table 1
xx P(X = x)P(X = x)
0 P(X=0)=250P(X=0)=250
1 P(X=1)=1150P(X=1)=1150
2 P(X=2)=2350P(X=2)=2350
3 P(X=3)=950P(X=3)=950
4 P(X=4)=450P(X=4)=450
5 P(X=5)=150P(X=5)=150

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

  1. Each P(X = x)P(X = x) is between 0 and 1, inclusive.
  2. The sum of the probabilities is 1, that is,

2 50 + 11 50 + 23 50 + 9 50 + 4 50 + 1 50 = 1 2 50 + 11 50 + 23 50 + 9 50 + 4 50 + 1 50 =1(1)

Example 2

Suppose Nancy has classes 3 days a week. She attends classes 3 days a week 80% of the time, 2 days 15% of the time, 1 day 4% of the time, and no days 1% of the time.

Problem 1

Let XX = the number of days Nancy ____________________ .

Solution

Let XX = the number of days Nancy attends class per week.

Problem 2

XX takes on what values?

Solution

0, 1, 2, and 3

Problem 3

Construct a probability distribution table (called a PDFPDF table) like the one in the previous example. The table should have two columns labeled xx and P(X = x)P(X = x). What does the P(X = x)P(X = x) column sum to?

Solution

Table 2
xx P(X = x)P(X = x)
0 0.01
1 0.04
2 0.15
3 0.80

Glossary

Probability Distribution Function (PDF):
A mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) , or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome.

Example:

A biased coin with probability 0.7 for a head (in one toss of the coin) is tossed 5 times. We are interested in the number of heads (the RV XX = the number of heads). XX is Binomial, so X ∼ B ( 5 , 0 . 7 ) X ∼ B ( 5 , 0 . 7 ) and P ( X = x ) =P(X=x)= 5 x . 7 x . 3 5 − x 5 x . 7 x . 3 5 − x or in the form of the table:

Table 3
xx P ( X = x ) P(X=x)
0 0.0024
1 0.0284
2 0.1323
3 0.3087
4 0.3602
5 0.1681

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