What is the Beta Function or Beta distribution?

Question

How can it be used to calculate the binomial distribution?

Answer 1

the Beta function is

Β(α, β) = ∫ x ^ (α - 1) (1 - x) ^ (β - 1) dx

the beta function is related to the Gamma function via:

Β(α, β) = Γ(α) Γ(β) / Γ(α + β)

where the limits of integration are from 0 to 1.

The Beta distribution has the probability density function:

f(x | α, β) = 1/ Β(α, β) * x ^ (α - 1) (1 - x) ^ (β - 1)

where 0 < x < 1; α > 0; β > 0

The Beta function is "used to model proportions, which naturally lie between 0 and 1"

if α = β = 1, then the distribution simplifies to the uniform distribution. The beta distribution is also related to the F distribution.

How is the Beta related to the binomial....

One method for generalizing the Binomial distribution is to allow the success probability to vary according to a distribution. Since the Beta distribution is useful for modeling continuous values between 0 and 1, we can use the beta distribution to vary the success probability of the binomial.

X ~ Binomial(n, p)
p ~ Beta(α, β)

to find the expectation and the variance of X in this case we need to use to different identities:

(1) E(X) = E(E(X|p))
(2) Var(X) = E(Var{X|p)) + Var(E(X|p))

the first identity says the expectation of X is equall to the expectation of the expectation of X given p;
the second identity says the variance of X is equall to the sum of the expectation of the variance of X given p and the variance of the expectation of X given p.

using these identities you can find:

E(X) = n * α / (α + β)
Var(X) = n * αβ (α + β + n) / ( (α + β)² * (α + β + 1))