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The gamma distribution generalises the exponential to k sequential events, with mean kθ and variance kθ². Its CDF is the regularised lower incomplete gamma function P(k, x/θ).
Gamma density
f(x) = x^(k−1)·e^(−x/θ) / (Γ(k)·θᵏ)
Mean and variance
E = kθ, Var = kθ²
P(X ≤ 5) ≈ 0.4562, with a mean of 6 and variance of 12.
k = 1 gives the exponential, and k = df/2 with θ = 2 gives the chi-square distribution.