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DISTRIBUTIONS / CONTINUOUS / EXPONENTIAL

Exponential distribution

What it describes

The Exponential distribution models the time to the next event in a Poisson process with constant rate. Its defining feature is memorylessness: elapsed waiting time does not change the distribution of the remaining wait.

It is positive and decreases from zero. lambda is a rate, not a scale: increasing the rate shortens typical waiting times and changes the natural unit of interpretation.

History and names

Its form appears in queueing theory and early models of calls and arrivals. It is the simplest continuous waiting-time model with constant instantaneous hazard.

How it connects to other distributions

It is Gamma with shape 1 and Erlang with k=1. Adding exponential waits gives Erlang; counting events in fixed time gives Poisson. It is also the only continuous lifetime distribution with constant hazard.

Where it appears

  • interarrival times in Poisson processes
  • lifetimes or waits with constant instantaneous risk

Fitting cautions

An exponential fit may look acceptable in a histogram but fail in its hazard rate; inspect survival or hazard when exposure time matters.

A clock that does not age

Suppose a component has already operated for one hundred hours. Under an Exponential model, its remaining lifetime has the same distribution it had when new. This memoryless property does not mean the component forgets; it means its instantaneous failure rate remains constant.

The property fits the next arrival in a homogeneous Poisson process and some decay mechanisms, but it rarely describes the complete life cycle of a machine. Wear creates increasing risk, while early defects can create decreasing risk. Weibull allows both patterns. Exponential remains a powerful benchmark: when it fails, the direction of departure tells a story about ageing, selection, or a rate that changes over time.

Decision guide

A good candidate when: time to the next event has a constant hazard and the process retains no memory of how long it has already waited.

Compare it with: Weibull and Gamma. A systematic trend in the hazard or exponential survival plot indicates that the memoryless assumption is too rigid.

References

  • SciPy reference: scipy.stats.expon — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.
  • Erlang, A. K. (1909). The theory of probabilities and telephone conversations. Nyt Tidsskrift for Matematik B, 20, 33–39.

Exponential Distribution: equations and calculator

Distribution defintion

X∼Exponential(λ)X\sim\mathrm{Exponential}\left(\lambda\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

λ∈R+\lambda\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−e−λxF_{X}\left(x\right)=1-e^{-\lambda x}

Probability density function

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

Percent point function/Sample

FX−1(u)=−ln⁡(1−u)λF^{-1}_{X}\left(u\right)=-\frac{\ln(1-u)}{\lambda}

Non-central parametric moments

μk′=E[Xk]=∫0∞xnfX(x)dx=k!λk\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{n}f_{X}\left(x\right)dx=\frac{k!}{\lambda^{k}}

Parametric mean

Mean(X)=μ1′=1λ\mathrm{Mean}(X)=\mu'_{1}=\frac{1}{\lambda}

Parametric variance

Variance(X)=μ2′−μ1′2=1λ2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{1}{\lambda^2}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=2

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=9\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=9

Parametric median

Median(X)=ln⁡2λ\mathrm{Median}(X)=\frac{\ln 2}{\lambda}

Parametric mode

Mode(X)=0\mathrm{Mode}(X)=0

Additional information and definitions

λ:Inverse of scale parameter\lambda:\text{Inverse of scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}