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Binomial Distribution

Formula

\[ X\sim \mathrm{Binomial}(n,p),\qquad P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} \]

Plot

type: bars
xs: 0 | 1 | 2 | 3 | 4 | 5
ys: 0.03125 | 0.15625 | 0.3125 | 0.3125 | 0.15625 | 0.03125
xmin: -0.5
xmax: 5.5
ymin: 0
ymax: 0.36
height: 280
title: Binomial PMF example (n=5, p=0.5)

Parameters

  • \(n\): number of trials
  • \(p\): success probability per trial
  • \(k\): number of successes

What it means

Models the number of successes in \(n\) independent Bernoulli trials with the same success probability.

What it's used for

  • Count outcomes from repeated yes/no trials.
  • Baseline model for proportions.

Key properties

  • Mean \(np\), variance \(np(1-p)\).
  • Support \(k=0,1,\dots,n\).

Common gotchas

  • Requires independent trials with constant \(p\).
  • Poisson is only an approximation in a specific rare-event regime.

Example

Number of heads in 10 fair coin flips is \(\mathrm{Binomial}(10,0.5)\).

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

See also