Beta Distribution

A distribution similar to the binomial, but where the probability of success $p$ is not a constant

Probability Distribution Function

A random variable $X$ has a beta distribution if and only if it has the pdf

f (x; α, β) = \frac{Γ (α + β)}{Γ (α) Γ (β)} x^{α - 1} (1 - x)^{β - 1}

for $0 < x < 1, α > 0, β > 0$

Beta is the continuous Binomial Distribution, where the combination is replaced by the continuous factorials in the Gamma function

(\binom{n}{x}) = \frac{n!}{(n - x)! x!} = \frac{(n - x + x)!}{(n - x)! x!} =

\int_{0}^{1} f (x; α, β) = \int

Mean

\begin{aligned} E (X^{r}) & = \int_{0}^{1} x^{r} \frac{Γ (α + β)}{Γ (α) Γ (β)} x^{α - 1} (1 - x)^{β - 1} d x \\ = \frac{Γ (α + β)}{Γ (α) Γ (β)} \frac{Γ (r + α) Γ (β)}{Γ (r + α + β)} \int_{0}^{1} \frac{Γ (r + α + β)}{Γ (r + α) Γ (β)} x^{r + α - 1} (1 - x)^{β - 1} d x \\ = \frac{Γ (α + β)}{Γ (α) Γ (β)} \frac{Γ (r + α) Γ (β)}{Γ (r + α + β)} \cdot 1 \\ = \frac{Γ (α + β)}{Γ (α + β + r)} \frac{Γ (r + α)}{Γ (α)} \\ = \frac{(α + r - 1) (α + r - 2) \dots α}{(α + β + r - 1) (α + β + r - 2) \dots (α + β)}, r = 1, 2, 3 \dots \end{aligned}

Uniform Distribution is a special case of the beta distribution with $α = 1$ and $β = 1$

X \sim u n i f (0, 1) \equiv B e t a (1, 1)

f (x) = \frac{Γ (α + β)}{Γ (α) Γ (β)} x^{α - 1} (1 - x)^{β - 1} = \frac{Γ (2)}{Γ (1) Γ (1)} x^{0} (1 - x)^{0} = 1, 0 < x < 1