Chapter 5: Discrete Probability Distributions - Poisson Distribution

Overview

• Discrete Probability Distributions: Deal with discrete variables, i.e., whole numbers or countable events.
• Poisson Distribution: Focuses on the number of outcomes occurring in a given time interval or specified region.
  • Suitable for events that are countable and occur independently.

Key Concepts

• Independence: Events must be independent; one event does not affect the probability of another event occurring (memoryless property).
• Average Rate (Lambda, ( \lambda )): Constant average rate of occurrence.
  • ( \lambda = ) average number of outcomes per unit time.
• Mean and Variance: Both are equal to ( \lambda T ).
  • ( \sigma = \sqrt{\lambda T} )

Poisson Distribution Formula

• Probability Distribution Formula: Used to calculate the exact probability of a given number of outcomes.
  • Formula: ( P(X) = \frac{e^{-\lambda T} (\lambda T)^x}{x!} )
  • E (Euler's Number): A mathematical constant approximately equal to 2.71828.

Examples of Poisson Variables

• Events per time interval:
  • Number of machines down in a shift.
  • Shipments received per day.
  • Spam calls in an hour.
  • Car accidents in a month.
• Events per specified region:
  • Number of field mice per acre.
  • Bacteria in a culture.
  • Typing errors per page.

Example Problem

• Scenario: Average number of radioactive particles passing through a counter in one millisecond is 4.
  • ( \lambda = 4 ) particles/ms.
• Problem: Probability that 6 particles enter the counter in any given millisecond.
  • Use ( P(X=6) = \frac{e^{-4} (4)^6}{6!} )
  • Result: Probability ( \approx 0.1042 ).

Using Poisson Distribution Table

• Similar to binomial tables, gives cumulative probabilities.
• Big F (Cumulative Probability): ( F(x) = \sum f(x) ) from 0 to x.
  • Procedure:
    • Find ( F(6) ) and ( F(5) ).
    • Calculate ( f(6) = F(6) - F(5) ).
  • Example: ( F(6) - F(5) = 0.8893 - 0.7851 = 0.1042 ).

Summary

• The Poisson distribution is useful for modeling the probability of a given number of events happening in a fixed interval of time or space.
• Understanding and using both the formula and distribution tables can provide flexibility in solving probability problems related to discrete events.