Exponential distribution

The exponential 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;τ)=1τe−x/τf_{X}(x;\tau)=\frac{1}{\tau}e^{-x/\tau}

where τ\tau is a positive real scale parameter. Equivalently, it can be parameterized as

fX(x;λ)=λe−λxf_{X}(x;\lambda)=\lambda e^{-\lambda x}

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

This distribution is commonly used in physics to model the wait times of random independent events, such a radioactive or particle decay. τ\tau is then usually interpreted as the characteristic time of the process. See the radioactive decay law for a related phenomenon.

Moments

The raw and central moment-generating functions are

MX∗(t)=11−tτ,MX(t)=e−tτ1−tτM^{*}_{X}(t)=\frac{1}{1-t\tau},\qquad M_{X}(t)=\frac{e^{-t\tau}}{1-t\tau}

Some moments are:

  • Raw 0. μ0∗=1\mu_{0}^{*}=1
    1. μ1∗=τ\mu_{1}^{*}=\tau (Expected value)
  • Central 0. μ0=1\mu_{0}=1
    1. μ1=0\mu_{1}=0
    2. μ2=τ2\mu_{2}=\tau ^{2} (Variance)
    3. μ3=2τ3\mu_{3}=2\tau ^{3}
    4. μ4=9τ4\mu_{4}=9\tau^{4}
  • Coefficients
    1. γ1=2\gamma_{1}=2 (skewness, it asymmetrical around the mean)
    2. γ2=6\gamma_{2}=6 (kurtosis)

Relation to other distributions

  • The sum of kk iid exponential random variables follows an Erlang distribution Erlang(x;k,τ)\text{Erlang}(x;k,\tau).