Erlang distribution

The Erlang distribution is a continuous Probability distribution over non-negative reals [0,∞)[0,\infty). For a Random variable XX, the Probability density function is

fX(x;k,τ)=1(k−1)!1τkxk−1e−x/τf_{X}(x;k,\tau)=\frac{1}{(k-1)!} \frac{1}{\tau^{k}}x^{k-1}e^{-x/\tau}

where k≥1k\geq 1 is an integer shape parameter and τ\tau is a positive real scale parameter. Equivalently, it can be parameterized as

fX(x;k,λ)=1(k−1)!λkxk−1e−λxf_{X}(x;k,\lambda)=\frac{1}{(k-1)!} \lambda^{k}x^{k-1}e^{-\lambda x}

where λ\lambda is a positive real rate parameter. The two are related by λ=1/τ\lambda=1/\tau.

It is often used to model the time it takes for kk Poisson-distributed events to occur (in a Poisson process). In this sense, it is the "inverse" of the Poisson distribution, which instead models the number of events in a given time.

The μ=1\mu=1 is also commonly encountered. The expression for this case is

fT(t)=1(k−1)!tk−1e−tf_{T}(t)=\frac{1}{(k-1)!}t^{k-1}e^{-t}

Moments

The Expected value and Variance are

E[X]=kτ=kλ,var(X)=kτ2=kλ2\text{E}[X]=k\tau=\frac{k}{\lambda},\qquad\text{var}(X)=k\tau ^{2}=\frac{k}{\lambda ^{2}}

Relation to other distributions

  • For k=1k=1, we get an Exponential distribution, Exp(x;τ)\text{Exp}(x;\tau). In fact, an Erlang distribution is the sum of kk exponential random variables.
  • It is a special case of the Gamma distribution with integer α=k\alpha=k.