Triangular Probability Distribution
Triangular Probability Distributions
Triangular(a, b, M) refers to a probability density function (PDF) where the probability to the left of a or to the right of b is 0. M is the mode of the distribution, and it must be between a and b, inclusive. The total area under any PDF curve (and above the x-axis) is always 1. In a triangular distribution, this results in the area of a triangular region, hence the name. Because of this the highest point is (M, 2/(b-a)). The graph of the Triangular PDF consists of two rays on the x-axis on both ends and the important part in the middle is one or two line segments. It is two line segments when a < M < b, and it is one line segment when a = M (negative slope) or b = M (negative slope).
Triangular distributions are not typically used for many real-world applications, but they can be a useful tool for developing understanding of probability as areas under the PDF curve, where the areas are easily computed by elementary school geometry formulas. In this case, all areas can be computed by finding areas of triangle and possibly adding or subtracting them. Recall that the area of a triangle is A = (1/2)(base)(height). Here the base is a length on the x-axis, and the height is a vertical distance from the x-axis to the PDF curve.