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DISTRIBUTIONS / CONTINUOUS / EXPONENTIAL 2P

Exponential 2P distribution

Portrait

The two-parameter Exponential distribution adds location to an exponential waiting time. Random waiting begins after a threshold and then keeps a constant hazard rate.

Historical trail

The 2P form is a software convention, not a new memoryless law. It is used in reliability and survival analysis when an event cannot occur during an initial period.

Two useful connections

  • With loc=0 it becomes exponential. It is a shifted shape-one Gamma and contrasts with three-parameter Weibull, where risk can change with time in addition to a threshold.
  • Memorylessness holds only after the threshold: X-loc is the exponential component, not X measured from time zero.

A use case

It may suit a machine that cannot fail during a known start-up period and then has constant risk. If early and late failures differ, one location shift is not enough.

Diagnostic advice

Do not confuse a minimum observation time with a statistical location; left censoring can produce a similar appearance.

Memorylessness, but only beyond the threshold

The two-parameter version adds a constant loc to an Exponential waiting time. It can represent a minimum travel time before random arrivals begin or a known instrument latency.

Memorylessness belongs to the excess above the threshold. Before an observation reaches loc, the Exponential clock has not even started in this construction. A shift should not be confused with truncating an Exponential process already under way. The former changes the mechanism’s origin; the latter conditions which observations can be seen. When loc is estimated from the same data, the sample minimum has disproportionate influence and its uncertainty should be reported.

Decision guide

A good candidate when: a known minimum delay exists and the additional waiting time after it can be treated as memoryless Exponential.

Compare it with: Exponential with a fixed origin and 3P Weibull. Estimating loc from the sample minimum can create false precision about the threshold.

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.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.
  • Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed. Wiley.

Exponential 2P Distribution: equations and calculator

Distribution defintion

X∼Exponential2P(λ,Loc)X\sim\mathrm{Exponential_{2P}}\left(\lambda,\text{Loc}\right)

Distribution domain

x∈[Loc,∞)x\in\left[\text{Loc},\infty\right)

Parameters domain and parameters constraints

λ∈R+,Loc∈R\lambda\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Cumulative distribution function

FX(x)=1−e−λ(x−Loc)F_{X}\left(x\right)=1-e^{-\lambda (x-\text{Loc})}

Probability density function

fX(x)=λe−λ(x−Loc)f_{X}\left(x\right)=\lambda e^{-\lambda (x-\text{Loc})}

Percent point function/Sample

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

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=k!λk\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{k!}{\lambda^{k}}

Parametric mean

Mean(X)=Loc+μ~1′=Loc+1λ\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}=\text{Loc}+\frac{1}{\lambda}

Parametric variance

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

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=2

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=9\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=9

Parametric median

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

Parametric mode

Mode(X)=Loc\mathrm{Mode}(X)=\text{Loc}

Additional information and definitions

X~∼Exponential(λ)\tilde{X}\sim\mathrm{Exponential}\left(\lambda\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
λ:Inverse of scale parameter\lambda:\text{Inverse of scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}