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
Less Than
0
At Most
0
More Than
0
At Least
0
Based on formula
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