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DISTRIBUTIONS / CONTINUOUS / RECIPROCAL

Reciprocal distribution

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The Reciprocal distribution is uniform on the logarithmic scale: equal-ratio intervals receive equal probability, not equal-length intervals. It suits order-of-magnitude uncertainty between positive bounds.

Feature Reading
Support Support is [a,b] with 0 < a < b. Density falls as 1/x; on a logarithmic axis it becomes uniform.
Shape If Y is uniform on [log a, log b], then exp(Y) has this distribution. It is a bounded log-uniform law and contrasts with Uniform, which is flat on the original scale.

Origin and terminology

The idea appears in error theory and in choosing numbers when only order of magnitude is known. It also relates to Jeffreys’ logarithmic scale in Bayesian inference, although a prior requires an additional decision.

In real models

  • simulation across orders of magnitude
  • multiplicative uncertainty in physical, financial, or engineering parameters

Comparisons

Never allow a ≤ 0; a logarithmic scale alone does not justify a reciprocal law if the phenomenon is not multiplicative.

Uniformity depends on which scale you inspect

Reciprocal is Uniform after taking logarithms. It gives the same probability to one through ten as to ten through one hundred whenever both ranges fit inside its bounds, because ratios rather than additive distances matter.

It is also called log-uniform. Density proportional to 1/x connects with scale invariance and the Jeffreys prior for certain parameters. Finite bounds are essential for normalization. Choosing them decides how many orders of magnitude are plausible and often matters more than the interior shape. Changing units preserves ratios, but both bounds must be transformed consistently.

Decision guide

A good candidate when: positive values lie between known bounds and each multiplicative interval—for example each decade—should receive equal probability.

Compare it with: Uniform and Lognormal. The distribution is uniform on the logarithmic scale, so conclusions are highly sensitive to chosen bounds and consistent units.

References

  • SciPy reference: scipy.stats.loguniform — 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.
  • Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society A, 186, 453–461.

Reciprocal Distribution: equations and calculator

Distribution defintion

X∼Reciprocal(a,b)X\sim\mathrm{Reciprocal}\left(a,b\right)

Distribution domain

x∈[a,b]x\in\left[a,b\right]

Parameters domain and parameters constraints

a∈R+,b∈R+,a<ba\in\mathbb{R}^{+},b\in\mathbb{R}^{+},a < b

Cumulative distribution function

FX(x)=ln⁡(x)−ln⁡(a)ln⁡(b)−ln⁡(a)F_{X}\left(x\right)=\frac{\ln(x)-\ln(a)}{\ln(b)-\ln(a)}

Probability density function

fX(x)=1x(ln⁡(b)−ln⁡(a))f_{X}\left(x\right)=\frac{1}{x\left(\ln(b)-\ln(a)\right)}

Percent point function/Sample

FX−1(u)=exp⁡(ln⁡(a)+u×(ln⁡(b)−ln⁡(a)))F^{-1}_{X}\left(u\right)=\exp(\ln(a)+u\times \left(\ln(b)-\ln(a)\right))

Non-central parametric moments

μk′=E[Xk]=∫abxkfX(x)dx=bk−akk(ln⁡(b)−ln⁡(a))\mu'_{k}=E[X^k]=\int_{a }^{b}x^{k}f_{X}\left(x\right)dx=\frac{b^k-a^k}{k\left(\ln(b)-\ln(a)\right)}

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

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

Parametric skewness

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

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\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}}

Parametric median

Median(X)=exp⁡[ln⁡(a)+(ln⁡(b)−ln⁡(a))2]\mathrm{Median}(X)=\exp\left[\ln(a)+\frac{\left(\ln(b)-\ln(a)\right)}{2}\right]

Parametric mode

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

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}