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The beta-binomial(n, α, β) distribution is a discrete distribution generated by choosing the probability p for a binomial(n, p) distribution from a beta(α, β) distribution. The number of successes is a beta-binomial(n, α, β) random variable. The function B in the pmf denotes the Beta function.

Parameter Range Description
n n = 1, 2, ... Number of trials
α α > 0 Shape parameter
β β > 0 Shape parameter

Probability Mass Function

P ( X = x | n , α , β ) = ( n x ) B ( x + α , n − x + β ) B ( α , β )

Support

x = 0 , … , n

Mean

Variance

Example n α β
The probability each year that an apple contains a worm is a beta(0.5, 8) random variable. 100 apples are picked one year. Let X be the number containing worms. 100 0.5000 8.000
The probability that a random student can answer an exam question correctly has a beta(3, 2) distribution. Let X be the number of correct answers on a 20 question exam. 20 3.000 2.000
The probability that an archer in the Night's Watch can hit a white walker has a beta(10, 3) distribution. A randomly chosen archer fires 36 arrows. Let X be the number of white walkers hit. 36 10.00 3.000

X ~ Beta-Binomial(n, α, β)

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

E(X) = , Var(X) =

Note that the variance is the product of two terms:

  • nαβ/(α + β)² is the variance for a binomial distribution with the same expected value as the beta-binomial distribution.
  • (α + β + n)/(α + β + 1) is a multiplier greater than 1 for n > 1.

So, a beta-binomial distribution with n > 1 always has greater variance than a binomial distribution with the same expected value and number of trials.

The illustration above shows a set of n independent Bernoulli(p) trials, where p is chosen at random from a beta(α, β) distribution. The number of successes has a beta-binomial(n, α, β) distribution. This is similar to a binomial distribution, with the difference being that p is randomly chosen from a beta distribution instead of being fixed.

The simulation above shows a set of n independent Bernoulli(p) trials, where p is chosen at random from a beta(α, β) distribution. Successful trials are shown in green and failures in grey. The number of successes has a beta-binomial(n, α, β) distribution. The histogram accumulates the results of each simulation.

Y = Beta-Binomial(1, α, β) ~ Bernoulli(α/(α + β)) limα+β→∞ X ~ Binomial(n, α/(α + β)) Beta-Binomial(n, 1, 1) ~ Discrete Uniform(0, n)

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

E(Y) = , Var(Y) =

When n = 1, the beta-binomial is simply a Bernoulli distribution with p chosen from a beta(α, β) distribution, which has mean α/(α + β).

The reason this occurs is that as α + β → ∞, the variance of the beta distribution approaches zero. The probability p selected from the beta distribution then converges to the value α/(α + β).

The red lines show the beta-binomial distribution superimposed over the limiting binomial distribution.

Note that when α = β = 1, the beta(α, β) distribution is a uniform(0, 1) distribution. The beta-binomial distribution is then an average of all binomial(n, p) distributions, with 0 ≤ p ≤ 1.