The uniform distribution is a continuous distribution that assigns only positive probabilities within a specified interval (a, b) — that is, all values between a and b. (a and b are two constants; they may be negative or positive.)
A continuous distribution can't be illustrated with a histogram, because this would require an infinite number of bars. Instead, a continuous distribution may be illustrated with a line or a curve. Areas under the line or the curve correspond to probabilities.
With the uniform distribution, all values over an interval (a, b) are equally likely to occur. As a result, the graph that illustrates this distribution is a rectangle. The figure shows the uniform distribution defined over the interval (0, 10).
The horizontal axis shows the range of values for X (0 to 10). The distribution assigns a probability of 0 to any value of X outside of the interval from 0 to 10.
The width of this interval equals the upper limit (b) minus the lower limit (a), which equals b – a. So in the figure, the width equals 10 – 0 = 10. The width of this interval represents the base of the rectangle. The height of the rectangle equals 1 divided by the base (1/10 in this case). The height always equals 1 divided by the base; this ensures that the area of the rectangle always equals 1. Areas under this rectangle represent probabilities. The total probability for any distribution is 1; therefore, the area under the rectangle must equal 1.
The area of a rectangle equals the base times the height, or in mathematical terms,
