Definitions

Definition

A uniform distribution is probability distribution in which every value of the random variable has the same chance of occurring.

Below is the graph of a uniform distribution on the interval [0, 5].

Note that area under the curve in the graph above (actually inside the rectangle) is one square unit. (A = LW = (5-0)(0.2) = (5)(0.2) = 1). This graph is an example of a density curve.

Definition .

The graph of a continuous probability distribution is called a density curve. It must satisfy the following:

1. The total area under the curve is 1.

2. Each point of the curve must be on or above the horizontal axis. (i.e each point on the curve has a nonnegative y value.)

Example

For the uniform distribution on [0, 5], find

a.) the probability that x is between 1 and 4.

b.) the probability that x is greater than or equal to 4.

Solution .

For part a we must find the area pictured below.

The area of the rectangle under the curve from x =1 to x = 4 is found by

LW = (4-1)(0.2) = (3)(0.2) = 0.6. The probability that x is between 1 and 4 is then 0.6.

(Note that what we used here for W is actually the height of the rectangle.)

For part b, we must find the area under the curve to the right of x = 4 as pictured below.

The area needed is found by

LW = (5-4)(0.2) = (1)(0.2) = 0.2.