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Geometric distribution

From phenomenon to model

It is one of the earliest waiting laws in probability and arises naturally from Bernoulli and Pascal’s work. Modern uses include reliability, queues, and count processes.

What to look at

It is a positive count and p is the success probability per trial. Its defining feature is memorylessness in discrete time.

Useful relationships

It is time to the first success; Negative Binomial is time to r successes. Small p creates a long tail, while mixtures of success probabilities create overdispersed waiting counts.

Names you may also encounter

It is also called the geometric distribution or Pascal waiting distribution. Some texts count failures before success on {0,1,…}; that is a different convention.

When it makes sense

Memorylessness allows process restart without changing the law: under independence and constant p, previous failures do not alter future waiting.

What not to assume

Check support before interpreting p: the same name can have mean 1/p or one unit less depending on convention.

Two clocks with the same name

Two legitimate conventions coexist. One clock counts trials up to and including the first success, so it starts at one. The other counts failures before that success, so it starts at zero. Phitter uses the first. This one-unit difference changes the mean, quantiles, and every operational interpretation even when the success probability is identical.

After accounting for the support convention, the Geometric distribution is the only memoryless discrete law on the nonnegative integers. Waiting through many attempts does not bring the next success any closer. That property can describe independent trials, but it becomes implausible with learning, wear, or fatigue. Adding several geometric waiting times produces a Negative Binomial variable, so the two distributions view the same sequence from complementary angles.

Decision guide

A good candidate when: the number of trials until the first success is counted and success probability remains constant.

Compare it with: Negative Binomial for waiting until several successes. Confirm whether the implementation counts trials from one or failures from zero before interpreting means and quantiles.

References

Geometric Distribution: equations and calculator

Distribution defintion

X∼Geometric(p)X\sim\mathrm{Geometric}\left(p\right)

Distribution domain

x∈N+≡{1,2,… }x\in\mathbb{N}^{+}\equiv \left\{1,2,\dots\right\}

Parameters domain and parameters constraints

p∈(0,1)⊆Rp\in\left(0,1\right)\subseteq\mathbb{R}

Cumulative distribution function

FX(x)=1−(1−p)⌊x⌋F_{X}\left(x\right)=1-(1 - p)^{\lfloor x\rfloor}

Probability mass function

fX(x)=(1−p)x−1pf_{X}\left(x\right)=(1 - p)^{x-1}p

Percent point function/Sample

FX−1(u)=⌈ln⁡(1−u)ln⁡(1−p)⌉F^{-1}_{X}\left(u\right)=\left\lceil{\frac{\ln{(1-u)}}{\ln{(1-p)}}}\right\rceil

Parametric centered moments

E[Xk]=μk′=∑x=0∞xkfX(x)=∑x=0∞(1−p)xp⋅xkE[X^k]=\mu'_{k}=\sum_{x=0}^{\infty}x^{k}f_{X}\left(x\right)=\sum_{x=0}^\infty (1-p)^x p\cdot x^k

Parametric mean

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

Parametric variance

Variance(X)=(μ2′−μ1′2)=1−pp2\mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=\frac{1-p}{p^2}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2−p1−p\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2-p}{\sqrt{1-p}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=9+p21−p\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+\frac{p^2}{1-p}

Parametric median

Median(X)=⌈−1log⁡2(1−p)⌉\mathrm{Median}(X)=\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil

Parametric mode

Mode(X)=1\mathrm{Mode}(X)=1

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
⌊x⌋:Floor function\lfloor{x}\rfloor: \text{Floor function}
⌈x⌉:Ceiling Function\lceil{x}\rceil: \text{Ceiling Function}