Business Statistics For Dummies
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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 uniform distribution defined over the interval (0, 10).
The uniform distribution defined over the interval (0, 10).

The width of this interval equals the upper limit (b) minus the lower limit (a), which equals ba. 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,


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Alan Anderson, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.

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