Geometric Distribution¶

Formula¶

\[ P(X=k)=(1-p)^{k-1}p,\quad k=1,2,\dots \]

Plot¶

type: bars
xs: 1 | 2 | 3 | 4 | 5 | 6
ys: 0.3 | 0.21 | 0.147 | 0.1029 | 0.07203 | 0.050421
xmin: 0.5
xmax: 6.5
ymin: 0
ymax: 0.34
height: 280
title: Geometric PMF example (p=0.3)

Parameters¶

\(p\): success probability per trial
\(X\): trial index of first success

What it means¶

Models how many Bernoulli trials are needed to get the first success.

What it's used for¶

Waiting-time models in discrete time.
Reliability and queueing toy models.

Key properties¶

Memoryless: \(P(X>m+n\mid X>m)=P(X>n)\).
Mean \(1/p\).

Common gotchas¶

Some texts define support starting at 0 (failures before first success) instead of 1.
Must state parameterization explicitly.

Example¶

If \(p=0.2\), the probability the first success occurs on trial 3 is \(0.8^2\cdot 0.2\).

How to Compute (Pseudocode)¶

Input: distribution parameters and query values (for PMF/PDF/CDF or moments)
Output: distribution quantities

validate parameters
for each query value x (or count k):
  evaluate the PMF/PDF/CDF formula from the card
optionally compute moments/statistics from known closed forms or by summation/integration
return the requested values

Complexity¶

Time: Typically \(O(q)\) for \(q\) query values once parameters are known (assuming constant-time formula evaluation per query)
Space: \(O(q)\) for output values (or \(O(1)\) for a single query)
Assumptions: Parameter estimation/fitting cost is excluded; numerical special-function evaluation can affect constants for some families