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

Mathematics and Statistics

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.

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,

$2 50 + 11 50 + 23 50 + 9 50 + 4 50 + 1 50 =1$

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 ____________________.

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

b. X takes on what values?

b. 0, 1, 2, and 3

c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in [link]. The table should have two columns labeled x and P(x). What does the P(x) column sum to?

c.

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

# Chapter 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.

# Homework

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given in [link].

 x P(x) 3 0.05 4 0.40 5 0.30 6 0.15 7 0.10
1. In words, define the random variable X.
2. What does it mean that the values zero, one, and two are not included for x in the PDF?
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