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The beta distribution (also called the beta distribution of the first kind) is a family of continuous probability distributions defined on [0, 1]. This distribution is similar to the binomial distribution, except where the binomial models the number of successes (x), the beta models the probability (p) of success.
Beta distribution of the second kind is another name for the beta prime distribution.
Beta Distribution PDF and CDF
The beta distribution is parameterized by two free parameters, alpha (α) and beta (β), which control the shape of the graph. The gives the notation for the distribution, Β (α, β), where α and β are real numbers. Other notation for the shape parameters includes (p, q) [1].
The probability density function (PDF)is
Where
- 1/B(α,β) is a normalizing constant to force the function integrate to 1,
- B(α,β) =
The graph of the beta density function can take on a variety of shapes. For example, if α < 1 and Β < 1, the graph will be a U shaped distribution, and if α = 1 and Β = 2, the graph is a straight line.
The cumulative distribution function (CDF) is the regularized incomplete beta function
Uses of the Beta Distribution
The beta distribution is used for a variety of applications for modeling the behavior of random variables limited to intervals of finite length. Uses include:
- Bayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric and negative binomial distributions.
- The Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem),
- Task duration modeling.
- Project/planning control systems like PERT and CPM.
- In project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time [1].
Excel Beta Distribution: BETA.DIST
Excel 2010 and later:
Excel 2010 and later uses the BETA.DIST function. If you’re using earlier versions, use BETADIST instead. Inputs are as follows [3]:
- A value where you want to evaluate the function.
- Alpha (α) and Beta (β), which determine the distribution’s shape.
- The lower and upper bound.
- “Cumulative” — a logical value (TRUE/FALSE) that determines the function’s form. TRUE returns the CDF and FALSE returns the PDF.
Example problem: Calculate a cumulative distribution function for a beta distribution at 0.5 with α = 9, β = 10, lower bound = 0 and upper bound = 1.
- Type the value where you want to evaluate the function in cell A1. For this example, type “.5” in cell A1.
- Type the value for alpha in cell A2 and then type the value for beta in cell A1. For this example, type “9” in cell A2 and then type “10” in cell A3.
- Type the lower bound in cell A4 and then type the upper bound in cell A5. For this example, type “0” in cell A4 and then type “1” in cell A5.
- Type the beta distribution function into cell A6. The format of the function is =BETA.DIST(value,alpha,beta,cumulative,lower bound,upper bound). For this example, type “=BETA.DIST(A1,A2,A3,TRUE,A4,A5)” into cell A6. The number and letter combinations refer to the cells (for example, Cell A1 contains the cell where we want to evaluate the function). Press “Enter” to see the result for the beta distribution, which is 0.592735.
References
[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.
[2] 3-Points Estimating
[3] Glen, S. (2013). Excel for Statistics. Independently published.