The gamma function generalizes the factorial to all positive real numbers via the integral Γ(a) = ∫₀^∞ x^a e^(-x) dx/x, satisfying the recursive identity Γ(a+1) = aΓ(a) and reducing to Γ(n) = (n-1)! for positive integers; normalizing this integral produces the Gamma(a, λ) probability distribution, a continuous distribution on the positive reals. The Gamma distribution's central role in probability theory stems from its connection to the Poisson process: the waiting time until the nth arrival in a Poisson process (the sum of n i.i.d. Exponential(λ) inter-arrival times) is distributed as Gamma(n, λ), a fact proved via moment generating functions, making the Gamma distribution the continuous-time analog of the negative binomial distribution.
The gamma function generalizes the factorial to all positive real numbers via the integral Γ(a) = ∫₀^∞ x^a e^(-x) dx/x, satisfying the recursive identity Γ(a+1) = aΓ(a) and reducing to Γ(n) = (n-1)! …