Poisson Probability Calculator Formula

Understand the math behind the poisson probability calculator. Each variable explained with a worked example.

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Formulas Used

P(X = k)

probability = (pow(lambda, k) * pow(e, -lambda)) / factorial(k)

Expected Value

expected = lambda

Standard Deviation

std_dev = sqrt(lambda)

Variables

VariableDescriptionDefault
lambdaAverage Rate (lambda)4
kNumber of Events (k)2

How It Works

How to Calculate Poisson Probability

Formula

P(X = k) = (lambda^k * e^(-lambda)) / k!

The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate lambda. Both the mean and variance equal lambda.

Worked Example

A call center receives an average of 4 calls per hour. What is the probability of exactly 2 calls in an hour?

lambda = 4k = 2
  1. 01lambda = 4, k = 2
  2. 02P(X=2) = (4^2 * e^(-4)) / 2!
  3. 03= (16 * 0.01832) / 2
  4. 04= 0.29305 / 2
  5. 05= 0.14653

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