# 24 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable

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:

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

### 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 *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*.

**Probability distribution table for Example 1**

x |
P(x) |
---|---|

0 | P(x = 0) = [latex]\frac{2}{50}[/latex] |

1 | P(x = 1) = [latex]\frac{11}{50}[/latex] |

2 | P(x = 2) = [latex]\frac{23}{50}[/latex] |

3 | P(x = 3) = [latex]\frac{9}{50}[/latex] |

4 | P(x = 4) = [latex]\frac{4}{50}[/latex] |

5 | P(x = 5) = [latex]\frac{1}{50}[/latex] |

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

- Each
*P*(*x*) is between zero and one, inclusive.

*P*(*x*= 0) = [latex]\frac{2}{50}[/latex] > 0

*P*(*x*= 1) = [latex]\frac{11}{50}[/latex]> 0

*P*(*x*= 2) = [latex]\frac{23}{50}[/latex] > 0

*P*(*x*= 3) = [latex]\frac{9}{50}[/latex] > 0

*P*(*x*= 4) = [latex]\frac{4}{50}[/latex] > 0

*P*(*x*= 5) = [latex]\frac{1}{50}[/latex] > 0 - The sum of the probabilities is one, that is,

P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)

=[latex]\frac{2}{50}[/latex] + [latex]\frac{11}{50}[/latex] + [latex]\frac{23}{50}[/latex] + [latex]\frac{9}{50}[/latex] + [latex]\frac{4}{50}[/latex] + [latex]\frac{1}{50}[/latex]

= 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.

Let *X* = the number of days Nancy has classes in a week.

*X*takes on what values?

## Show Answer

0 day, 1 day, 2 days, 3 days

*x*and

*P*(

*x*).

## Show Answer

X | P(X) |

0 | 1% = 0.01 |

1 | 4% = 0.04 |

2 | 15% = 0.15 |

3 | 80% = 0.80 |

*P*(

*x*) column sum to?

## Show Answer

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

## Example 2

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?

## Show Answer

X is the number of days Jeremiah attends basketball practice per week.

X takes on the values 0, 1, and 2.

Number of days Jeremiah attends basketball practice per week, X | P(X) |

0 | 2% = 0.02 |

1 | 8% = 0.08 |

2 | 90% = 0.90 |

## Concept Review

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

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

### Solutions to Try These:

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 |