Gamma & Beta functions

Gamma function

definition:

Γ (x) = \int_{0}^{\infty} e^{- t} t^{x - 1} d t (x > 0)

Γ (1) = 1

Γ (0.5) = \sqrt{π}

Γ (n + 1) = n Γ (n) = (n)!

Γ (α, β) = f (λ) = λ^{α - 1} e^{- β λ}

definition:

B (x, y) = \int_{0}^{1} t^{x - 1} (1 - t)^{y - 1} d t (x > 0, y > 0)

p (t | x, y) = b e t a (t | x, y) = t^{x - 1} (1 - t)^{y - 1} / B (x, y)

where B(x, y) is a beta function as the normalizer.

B (α, β) = \frac{Γ (α) Γ (β)}{Γ (α + β)}