Section 6.3

The Poisson Probability Distribution

Model the number of independent events occurring in a fixed interval of time or space.

1

The Poisson Process

A Poisson process counts the number of occurrences of an event over a period of time, area, distance, or any other interval.

Conditions for Poisson

  • 1
    Independence: Occurrences in one interval do not affect other non-overlapping intervals.
  • 2
    Constant Rate: The average rate of occurrence is constant for all intervals of the same size.
  • 3
    Rare Events: Determining the probability of 2 or more occurrences in a very small interval is negligible.
Notation
Average rate (occurrences per unit) Interval length (time, space, etc.) Number of occurrences (0, 1, 2...)

Examples

  • Number of emails arriving in 1 hour
  • Number of typos per page in a book
  • Cars passing a checkpoint in 5 mins
2

Poisson Probability Formula

Probability Mass Function

LogicLens: Deconstructing the Formula

Part 1: The Weight ()

This term scales with the expected number of occurrences () raised to the power of the actual count ().

Part 2: The Decay ()

This exponential term rapidly decreases probabilities for high values of , ensuring the sum of all probabilities equals 1.

3

Mean & Standard Deviation

Mean

Same as the expected count in the interval.

Standard Deviation

Square root of the mean (Unique property!)

4

Interactive Poisson Calculator

Poisson Probability Calculator

Occurrences per interval unit

Number of units (e.g., hours, sq meters)

Mean ()
Std Dev ()
Exact
0.00000
Less Than
0.00000
At Most
0.00000
More Than
0.00000
At Least
0.00000
Based on formula
Adaptive Engine

LogicLens Practice Suite

Log in to Access Adaptive Practice

Our AI engine generates unique practice problems based on your progress.