A uniform distribution U(a, b), also called a rectangular distribution, is defined by two parameters:
- minimum, a.
- maximum, b.
The area under the curve of the uniform distribution is always equal to 1. In the above graph, the area is:
A = l x h = 3 * 0.333… = 1.
Continuous and Discrete Uniform Distribution
The continuous uniform distribution is shaped like a rectangle – like the example above. The discrete uniform distribution is also rectangular shaped, but a series of dots represents a known, finite number of outcomes.
As an example, one roll of a die roll has six possible outcomes: 1,2,3,4,5, or 6. There is an equal probability for each number (1/6).
PDF and CDF
The general formula for the probability density function (pdf) is:
f(x) = 1/ (B-A) for A ≤ x ≤B.
- A, the location parameter, defines the center of the graph.
- B, the scale parameter, stretches the graph on the x-axis.
Note: A and B shouldn’t be confused with lowercase (a, b), which refer to the interval (min, max).
The uniform cumulative distribution function adds up all of the probabilities and plots a linear graph:
Expected Value &Variance
The expected value (or mean) of a uniform random variable X is:
E(X) = (1/2) (a + b)
E(X) = (b + a) / 2.
For example, with a = 2 and b = 4, the expected value is E(X) = (4 + 2) / 2 = 3.
The variance of a uniform random variable is:
Var(x) = (1/12)(b – a)2
For example, with a = 2 and b = 4, the variance is Var(x) = (1/12)(4 – 2)2 = 1/3.