How to use the POISSON.DIST() function in Excel

This function calculates probabilities for a Poisson-distributed random variable. The Poisson distribution is commonly used to predict the frequency of rare, independent events over a specific interval (e.g., call center arrivals per hour or tire failures per 100,000 miles).

Syntax:
POISSON.DIST(x ; mean ; cumulative)

Arguments:

• x (required): The number of events to evaluate.
• mean (required): The expected average number of events (λ).
• cumulative (required): A logical value determining the calculation type:
  • TRUE: Returns the cumulative probability (0 to *x* events).
  • FALSE: Returns the exact probability for exactly *x* events.

Background:
Named after Siméon Denis Poisson (1781–1840), this distribution models rare events in large populations where:

• Events are independent (e.g., radioactive decay, customer arrivals).
• The average event rate (mean) is known but the actual occurrences are sporadic.
• The probability of an event is proportional to the interval length.

Key Properties:

1. Approximates the binomial distribution for low-probability, high-sample scenarios.
2. Requires only the mean (λ) as a parameter (unlike the binomial distribution).
3. Assumes events are:
   • Rare within the interval.
   • Random and independent of prior events.

Formulas:

• Exact probability (cumulative = FALSE):

• Cumulative probability (cumulative = TRUE):

Example:
Scenario: A tire dealer observes an average of 4 damage incidents per 100,000 miles.

1. Question: What is the probability of exactly 3 incidents?
   • Inputs:
     • x = 3, mean = 4, cumulative = FALSE
   • Result: 19.54% (see Figure below).

1. Question: What is the probability of 0 to 3 incidents?
   • Inputs:
     • x = 3, mean = 4, cumulative = TRUE
   • Result: 43.35% (see Figure below).