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

## 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?

### Solution

a. Let *X* = the number of minutes a person must wait for a bus. *a* = 0 and *b* = 15.
*X* ~ *U*(0, 15). Write the probability density function. *f* (*x*) = $\frac{1}{15\text{}-\text{}0}$ = $\frac{1}{15}$
for 0 ≤ *x* ≤ 15.

Find *P* (*x* < 12.5). Draw a graph.

$$P(x<k)=(\text{base})(\text{height})=(12.5-0)\left(\frac{1}{15}\right)=0.8333$$

The probability a person waits less than 12.5 minutes is 0.8333.

### Problem

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

### Solution

b. *μ* = $\frac{a\text{}+\text{}b}{2}$
= $\frac{15\text{}+\text{}0}{2}$
= 7.5. On the average, a person must wait 7.5 minutes.

*σ* = $\sqrt{\frac{(b-a{)}^{2}}{12}}=\sqrt{\frac{(\mathrm{15}-0{)}^{2}}{12}}$ = 4.3. The Standard deviation is 4.3 minutes.

### Problem

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

## NOTE

This asks for the 90^{th} percentile.

### Solution

c. Find the 90^{th} percentile. Draw a graph. Let *k* = the 90^{th} percentile.

$P(x<k)=(\text{base})(\text{height})=(k-0)(\frac{1}{15})$

$0.90=\left(k\right)\left(\frac{1}{15}\right)$

$k=(0.90)(15)=13.5$

The 90^{th} percentile is 13.5 minutes. Ninety percent of the time, a person must wait at most 13.5 minutes.

## 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.

- 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 in a single season is between 480 and 500 hours?