MCPCalc

Poisson Probability Calculator

Compute exact and cumulative Poisson probabilities with distribution moments.

Scratchpad (not saved)

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What This Calculator Does

This calculator evaluates the Poisson model for counts of rare independent events over a fixed interval, reporting both exact and cumulative probabilities. It also returns the standard Poisson moments so you can judge how wide the count distribution is around the average rate.

It combines Average Rate λ, Events k to estimate P(X = k), P(X <= k), P(X >= k).

Formula & Method

Core equations: P(X=k)=e−λλkk!,P(X≤k)=∑j=0kP(X=j)P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!},\quad P(X\le k)=\sum_{j=0}^{k}P(X=j).

Notation used in the formulas: RR = P(X = k); x1x_{1} = Average Rate λ; x2x_{2} = Events k.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it when events are described by an average arrival rate rather than a fixed number of Bernoulli trials, such as calls per hour, defects per batch, or claims per day.

Reference Book

Introduction to Mathematical Statistics

Global Edition

Robert V. Hogg, Joseph McKean, and Allen T. Craig · Pearson

A standard reference for probability distributions, estimation, hypothesis testing, regression, and ANOVA.

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Inputs Used

  • Average Rate λ: Used directly in the calculation.
  • Events k: Used directly in the calculation.

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