Gamma distribution

The Gamma distribution is the continuous waiting time for k independent Poisson events, each arriving at rate λ. Set k=1 and you recover the Exponential distribution exactly.

f (x; k, λ) = \frac{λ ^{k} x ^{k - 1} e ^{- λ x}}{Γ ( k )}, x > 0

Mean = k/λ

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Variance = k/λ²

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Special case

Erlang(2)

k (shape) 2.0 λ (rate) 1.0 n (samples) 500

Solid curve:

Γ (k, λ)

PDF. At k=1 this is exactly Exponential(λ). Integer k gives an Erlang distribution — the waiting time for k Poisson events.

What to notice

k=1: reduces exactly to Exponential(λ) — a rapidly decaying curve.
Integer k (Erlang): the distribution of the sum of k independent Exp(λ) random variables. Modes and peaks become more pronounced.
Large k: the distribution becomes symmetric and bell-shaped — another manifestation of the Central Limit Theorem, since you’re summing k exponential random variables.
Rate λ stretches or compresses the x-axis without changing the shape. Mean = $k / λ$ , Variance = $k / λ^{2}$ .

Family connections

The Gamma family intersects much of probability theory:

k=1: Exponential(λ)
k=n/2, λ=1/2: Chi-squared(n) — used in statistical tests
k=α, λ=β: the conjugate prior for the Poisson rate parameter in Bayesian statistics
k → ∞: approaches Normal by CLT

Shape parameter k

For non-integer k the distribution is defined via the Gamma function Γ(k) — the continuous extension of the factorial: Γ(n) = (n−1)! for integer n. The k parameter controls how many “modes” worth of waiting you’re modeling. Fractional k arises naturally in Bayesian inference as a posterior distribution over rates.