The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints.

The mathematical statement of the uniform distribution is

*f*(*x*) = $\frac{1}{b-a}$ for *a* ≤ *x* ≤ *b*

where *a* = the lowest value of *x* and *b* = the highest value of *x*.

Formulas for the theoretical mean and standard deviation are

$\mu =\frac{a+b}{2}$ and $\sigma =\sqrt{\frac{{(b-a)}^{2}}{12}}$

The data that follow are the number of passengers on 35 different charter fishing boats. The sample mean = 7.9 and the sample standard deviation = 4.33. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. State the values of *a* and *b*. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation.

1 | 12 | 4 | 10 | 4 | 14 | 11 |

7 | 11 | 4 | 13 | 2 | 4 | 6 |

3 | 10 | 0 | 12 | 6 | 9 | 10 |

5 | 13 | 4 | 10 | 14 | 12 | 11 |

6 | 10 | 11 | 0 | 11 | 13 | 2 |

### Example 5.2

The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive.

a. What is the probability that a person waits fewer than 12.5 minutes?

b. On the average, how long must a person wait? Find the mean, *μ*, and the standard deviation, *σ*.

c. Ninety percent of the time, the time a person must wait falls below what value?

### NOTE

This asks for the 90^{th} percentile.

The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive.

- Find
*a*and*b*and describe what they represent. - Write the distribution.
- Find the mean and the standard deviation.
- What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours?