🎲 Probability Distributions

Continuous & discrete distributions, inverse CDFs, random generators, combinatorics. All commands accept a value to compute CDF/PMF, or x to return a function.

✓ 27 Implemented ⏳ 15 Coming Soon

Continuous Distributions — Implemented

Normal( <mu>, <sigma>, <v> )

Aliases: normal, NormalDist

✓ Done

Normal (Gaussian) distribution. With 3 args returns CDF P(X≤v). Optional 4th boolean: false returns PDF.

Examples

Normal(0, 1, 1.96)        → 0.975002  (CDF)
Normal(2, 0.5, 1)         → 0.022750  (CDF)
Normal(0, 1, 0, false)    → 0.398942  (PDF at 0)

InversNormal( <mu>, <sigma>, )

Aliases: inversenormal, InverseNormal

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Inverse normal CDF — returns x such that P(X≤x) = p.

Examples

InversNormal(0, 1, 0.975)   → 1.959964 (z-score)
InversNormal(100, 15, 0.9)  → 119.224  (IQ example)

TVerteilung( <df>, <v> )

Aliases: tdist, TDist

✓ Done

Student's t-distribution CDF with df degrees of freedom.

Examples

TVerteilung(10, 0)      → 0.5
TVerteilung(30, 1.96)   → 0.9702

InversTVerteilung( <df>, )

Aliases: inversetdist

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Inverse t-distribution. Returns t-critical value for given degrees of freedom and probability.

Examples

InversTVerteilung(10, 0.975)   → 2.2281  (95% CI)

ChiQuadratVerteilung( <df>, <v> )

Aliases: chisqdist

✓ Done

Chi-squared distribution CDF.

Examples

ChiQuadratVerteilung(4, 3)    → 0.4422
InversChiQuadrat(4, 0.95)     → 9.4877

FVerteilung( <df1>, <df2>, <v> )

Aliases: fdist, FDist

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F-distribution CDF with numerator df1 and denominator df2 degrees of freedom.

Example

FVerteilung(4, 20, 2.5)   → 0.9185

Exponentialverteilung( <lambda>, <v> )

Aliases: expdist, ExponentialDist

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Exponential distribution CDF: P(X≤v) = 1 - e^(-λv).

Example

Exponentialverteilung(2, 1)   → 0.8647

Gammaverteilung( <alpha>, <beta>, <v> )

✓ Done

Gamma distribution CDF. Shape parameter alpha, scale parameter beta.

Example

Gammaverteilung(2, 2, 4)   → 0.5940

Weibull( <k>, <lambda>, <v> )

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Weibull distribution CDF. Shape parameter k, scale parameter lambda.

Example

Weibull(0.5, 1, 1)   → 0.6321

Cauchy( <zentrum>, <breite>, <v> )

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Cauchy distribution CDF. Heavy-tailed distribution with location and scale parameters.

Example

Cauchy(1, 2, 3)   → 0.75

LogNormal( <mu>, <sigma>, <v> )

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Log-normal distribution CDF. x is log-normally distributed when ln(x) is normally distributed.

Example

LogNormal(0, 1, 2)   → 0.7558

Gleichverteilung( <a>, , <v> )

Aliases: uniform, UniformDist

✓ Done

Uniform distribution CDF on interval [a, b].

Example

Gleichverteilung(0, 10, 4)   → 0.4

Discrete Distributions — Implemented

Binomial( <n>, , <v>, <kumulativ?> )

Aliases: binomial, BinomialDist

✓ Done

Binomial distribution. false = P(X=v), true = P(X≤v). Default: PMF.

Examples

Binomial(10, 0.5, 5, false)   → 0.2461  P(X=5)
Binomial(10, 0.5, 5, true)    → 0.6230  P(X≤5)

Poisson( <lambda>, <v>, <kumulativ?> )

Aliases: poisson, PoissonDist

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Poisson distribution with mean lambda.

Examples

Poisson(3, 2, false)   → 0.2240  P(X=2)
Poisson(3, 2, true)    → 0.4232  P(X≤2)

Hypergeometrisch( <N>, <K>, <n>, <v>, <kumulativ?> )

Aliases: hypergeometric

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Hypergeometric distribution. Drawing without replacement: N total, K successes, n drawn.

Examples

Hypergeometrisch(10, 2, 2, 0, false)   → 0.6222
Hypergeometrisch(10, 2, 2, 1, true)    → 0.9778

Combinatorics — Implemented

BinomialKoeffizient( <n>, <r> )

Aliases: nCr, binomcoeff, Binomialkoeffizient

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"n choose r" — number of ways to choose r items from n. = n! / (r!·(n−r)!)

Examples

BinomialKoeffizient(10, 3)   → 120
BinomialKoeffizient(52, 5)   → 2598960 (poker hands)

Random Generators — Implemented

Zufallszahl( <min>, <max> )

Aliases: random, Random, RandomBetween

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Random integer uniformly distributed between min and max (inclusive).

Example

Zufallszahl(1, 6)    → random die roll
Zufallszahl(0, 100)  → random 0 to 100

ZufallszahlNormalverteilt( <mu>, <sigma> )

Aliases: randomnormal, RandomNormal

✓ Done

Random number from a normal distribution with given mean and standard deviation.

Example

ZufallszahlNormalverteilt(0, 1)    → e.g. -0.432
ZufallszahlNormalverteilt(170, 8)  → random height (cm)

ZufallszahlBinomialverteilt( <n>, ) / ZufallszahlPoissonverteilt( <lambda> )

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Random binomial or Poisson variate.

Example

ZufallszahlBinomialverteilt(10, 0.3)   → e.g. 3
ZufallszahlPoissonverteilt(5)          → e.g. 4

ZufaelligesElement( <Liste> ) / Stichprobe( <Liste>, <n> ) / Mischen( <Liste> )

✓ Done

Pick from list: random element, random sample of n, or shuffle entire list.

Example

ZufaelligesElement({1,2,3,4,5})      → e.g. 3
Stichprobe({1,2,3,4,5}, 3)           → e.g. {2, 5, 1}
Mischen({1,2,3,4,5})                 → e.g. {3, 1, 5, 2, 4}

Coming Soon

Dreiecksverteilung( <a>, , <modus>, <v> )

⏳ Soon

Triangle distribution on [a, b] with mode (peak) at modus. Useful when you know min, max, and most likely value.

Planned

Dreiecksverteilung(0, 5, 2, 2)   → 0.4

Erlang( <n>, <lambda>, <v> )

⏳ Soon

Erlang distribution — special case of Gamma with integer shape. Used in queuing theory.

LogistischeVerteilung( <mu>, <s>, <v> ) / InversLogistischeVerteilung

⏳ Soon

Logistic distribution — similar to normal but with heavier tails. Used in logistic regression.

NegativBinomial( <n>, , <v>, <kumulativ?> )

⏳ Soon

Negative binomial — number of failures before n-th success in Bernoulli trials.

Zeta( <n>, <exponent>, <v>, <kumulativ?> ) / InversZeta

⏳ Soon

Zeta (power law) distribution. Probability proportional to k^(-exponent).

Bernoulli( , <kumulativ?> )

⏳ Soon

Bernoulli distribution — single trial with success probability p. Special case of Binomial(1, p).

nPr( <n>, <r> )

⏳ Soon

Permutations — number of ordered arrangements: n!/(n−r)!

Planned

nPr(10, 2)   → 90

ZufallszahlGleichverteilt( <min>, <max> ) / ZufaelligesPolynom

⏳ Soon

Continuous uniform random float, and random polynomial with specified degree and coefficient range.

InversWeibull / InversLogNormal / InversGamma / InversHypergeometrisch

⏳ Soon

Inverse CDF functions for Weibull, LogNormal, Gamma, and Hypergeometric distributions.