Beta Density Function
A Beta distribution is a family of probability distributions that stretch from 0 to 1 on the number line. The letters α and β define the shape of the curve. The Beta distribution is an excellent way to represent outcomes like probabilities or proportions.
The values of α and β determine the shape of the beta density function. For example, if α < 1 and β < 1, the graph’s shape will be a “U” (see the red plot on the picture above, and if α = 1 and β = 2, the graph is a straight line; If you look at the graph above, the blue line is almost a straight line: that’s because α = 1 and β = 3.
The probability function P(x) and distribution function D(x) for the Beta Distribution are:
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| Beta Distribution |
It’s unlikely you’ll ever have to use those ugly equations, as most math software simplifies the math for you. For example, in Mathematical you can implement a beta distribution by typing:
Beta Distribution [alpha,beta] and in Microsoft Excel you can use Beta Dist(x,shape_a,shape_b,min,max).
The beta distribution is used for many applications, including Bayesian statistics, the Rule of Succession (a famous example being Pierre-Simon Laplace’s treatment of the sunrise problem), and Task duration modeling. the beta distribution is especially suited to project/planning control systems like PERT and CPM because the function is constrained by an interval with a minimum (0) and maximum (1) value.
Tip:
Don’t get confused by all those betas. In (typical) mathematical tomfoolery, there are three different betas:
- In “B(α, β),” Beta is the name of the function in the denominator of the density function.
- In “Beta(α, β),” Beta means the name of the probability distribution.
- In “β,” Beta is the name of the second parameter in the density function.
References:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 944-945, 1972
Evans, M.; Hastings, N.; and Peacock, B. “Beta Distribution.” Ch. 5 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 34-42, 2000
