Statistics How To

Beta Distribution: Definition, Calculation

Probability Distributions > Beta Distribution

What is a Beta Distribution?

A Beta distribution is a type of probability distribution. This distribution represents a family of probabilities and is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that Kanye West will win the next Presidential election? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives you a way to describe this.

One reason that this function is confusing is there are three “Betas” to contend with, and they all have different meanings:

  • Beta(α, β): the name of the probability distribution.
  • B(α, β ): the name of a function in the denominator of the pdf. This acts as a “normalizing constant” to ensure that the area under the curve of the pdf equals 1.
  • β: the name of the second shape parameter in the pdf.

The basic beta distribution is also called the beta distribution of the first kind. Beta distribution of the second kind is another name for the beta prime distribution.

Probability Density Function

The general formula for the probability density function is:
beta distribution


where:
α and β are two positive shape parameters which control the shape of the distribution.

beta distribution

The Beta Distribution pdf.



Software Options

Most software packages have options for the beta distribution.

Mathematica
Implement a beta distribution by typing BetaDistribution[alpha,beta].

R
dbeta(x, shape1, shape2, ncp = 0, log = FALSE)
pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)
rbeta(n, shape1, shape2, ncp = 0)
where:
x, q = vector of quantiles.
p = vector of probabilities.
n = # of observations.
shape1, shape2 = shape parameters α and β
ncp = non-centrality parameter.
log, log.p = logical; if TRUE, probabilities p are given as log(p).
lower.tail = logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].


Excel Beta Distribution

The Microsoft Excel beta distribution format depends on which version of Excel you are using; Microsoft Excel 2010 uses the BETA.DIST function, while earlier versions use BETADIST. You’ll need to know alpha and beta, both of which refer to the shape of the graph. Five inputs are needed for the beta distribution:

  • The value where you want to evaluate the function.
  • Alpha and Beta, the parameters of the distribution which determine shape.
  • The lower and upper bound.

An extra input is needed for the Excel 2010 Beta Distribution: cumulative. Cumulative is a logical value that determines the function’s form and can either be TRUE (returns the cumulative distribution function) or FALSE (returns the probability density function).

Watch the video or read the steps below:

Sample problem: Calculate a cumulative probability function for a beta distribution in Excel at 0.5 with an alpha of 9, a beta of 10, a lower bound of 0 and an upper bound of 1.

Excel 2003, Excel 2007:

Step 1: Type the value where you want to evaluate the function in cell A1. For this example, type “.5” in cell A1.

Step 2: 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.

Step 3: 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.

Step 4: Type the beta distribution function into cell A6. The format of the function is =BETADIST(value,alpha,beta,lower bound,upper bound). For this example, type “=BETADIST(A1,A2,A3,A4,A5)” into cell A6. Press “Enter” to see the result for the beta distribution, which is 0.592735.

Excel Beta Distribution

Excel 2010

Step 1: Follow steps 1 through 3 in the Excel 2003-2007 section above.

Step 2: 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. Press “Enter” to see the result for the beta distribution, which is 0.592735.

Beta Density Function

The Beta distribution is an excellent way to represent outcomes like probabilities or proportions.
beta density function
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:
beta PDF

Applications of the Beta Density Function

The beta distribution is used for many applications, including Bayesian hypothesis testing, 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:

  1. In “B(α, β),” Beta is the name of the function in the denominator of the density function.
  2. In “Beta(α, β),” Beta means the name of the probability distribution.
  3. Beta (β) 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.

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If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Beta Distribution: Definition, Calculation was last modified: October 12th, 2017 by Stephanie Glen

5 thoughts on “Beta Distribution: Definition, Calculation

  1. Vastavikta Gandhi

    Thank you very much for the article. Where can I find a few examples on problems on beta distribution with their solutions?

  2. itool

    Very useful!

    How would you explain the non centrality parameter (ncp) in R? What does the default value 0 mean?

    Thanks in advance!

  3. Andale Post author

    Itool. NCPs hurt my head, lol. You can get an overview of what the default value of 0 means here:
    What is an NCP?
    It’s basically saying the distribution is central.