# 5.2The Uniform Distribution

Introductory Business Statistics 2e5.2 The Uniform Distribution

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) = $1 b−a 1 b−a$ for axb

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

Formulas for the theoretical mean and standard deviation are

$μ= a+b 2 μ= a+b 2$ and $σ= (b−a) 2 12 σ= (b−a) 2 12$

## 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.

### Problem

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

### Problem

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

### Problem

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

## NOTE

This asks for the 90th percentile.

## Try It 5.2

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

1. Find a and b and describe what they represent.
2. Write the distribution.
3. Find the mean and the standard deviation.
4. What is the probability that the duration of games for a team in a single season is between 480 and 500 hours?