Click or press any key to close help Help screen describing the interactive elements

The beta distribution is a continuous distribution on the interval [0, 1]. It arises naturally as the distribution of the order statistics of a uniform random sample. If n points are randomly chosen from the interval (0, 1) and arranged in order, then the jth point has a beta(j, n - j + 1) distribution. The term B(α, β) in the denominator of the pdf denotes the Beta function.

Parameter Range Description
α α > 0 Shape parameter
β β > 0 Shape parameter

Probability Density Function

f ( x | α , β ) = 1 B ( α , β ) x α − 1 ( 1 − x ) β − 1

Support

0 ≤ x ≤ 1

Mean

Variance

Example α β
Five numbers are chosen at random from the interval (0, 1) and arranged in order. The middle number has a beta(3, 3) distribution. 3.000 3.000
Relative humidity for Auckland, New Zealand for the first six months of 2022 has an approximate beta(5.93, 1.78) distribution. 5.930 1.780
The probability that an apple will be unblemished depends on yearly weather and insect populations, and has a beta(8, 0.5) distribution. 8.000 0.5000

X ~ Beta(α, β)

Chart of the beta distribution Chart area for displaying the beta pdf, cdf, visualization, and simulation

E(X) = , Var(X) =

Because it is bounded between 0 and 1, this distribution is often used to model quantities representing fractions or percentages. The beta distribution is also used to model the distribution of probabilities, when the probability of an event is itself a random variable. It can take many shapes, either horizontal, strictly increasing or decreasing, u-shaped, or unimodal.

The illustration above shows a sample of n points chosen randomly and independently from the interval [0, 1]. The order statistics X(1), …, X(n) of this sample are the sample values placed in ascending order. The jth order statistic X(j) has a beta(j, n - j + 1) distribution.

The simulation above shows a set of n points chosen randomly from the interval [0, 1], where n = α + β - 1 after rounding α and β to the nearest positive integer. When the points are arranged in order, the location of point α has a beta(α, β) distribution. The histogram accumulates the results of each simulation.

Y = 1 - X ~ Beta(β, α) limα,β→∞ X ~ Normal(E(X), Var(X)) Beta(1, 1) ~ Standard Uniform

Chart of the related distribution Chart area for displaying the related pdf, cdf, visualization, and simulation

E(Y) = , Var(Y) =

This can be seen in the graph above, where swapping α and β gives a mirror image of the original beta distribution.

This convergence can be seen by reducing the variance in the graph of the beta distribution. The probability mass then becomes more concentrated around the mean, and the distribution becomes more bell-shaped.

The red dashed line shows the distribution of Beta(α, β), which can be seen to converge to a normal distribution as α, β → ∞.

Note that the jth point X(j) in an ordered uniformly random sample of size n has a beta(j, n - j + 1) distribution. A beta(1, 1) distribution therefore has j = n = 1, which is the distribution of a standard uniform point.