Day 10

Math 216: Statistical Thinking

Bastola

Introduction to Poisson Random Variables



Definition

A Poisson Random Variable is typically used to model the number of times an event occurs in a fixed interval of time or space. It is defined by the parameter \(\lambda\) (lambda), which represents the average number of events in the given interval. This distribution is particularly useful for modeling the frequency of rare events over a fixed period or in a specified area.

Introduction to Poisson Random Variables


  • Discrete Outcomes: Counts of events (0, 1, 2, …).
  • Rate of Occurrence: Governed by \(\lambda\), which is both the mean and the variance of the distribution.
  • Independence: The occurrence of one event does not affect the probability of another event occurring in the same interval.

Probability Distribution, Mean, and Variance of Poisson Random Variables

  • Formula for Probability:

\[\begin{align*} p(x)&=\frac{\lambda^x e^{-\lambda}}{x!}\quad(x=0,1,2,\ldots) \end{align*}\]

  • Mean and Variance:
    • Mean (\(\mu\)): \(\mu=\lambda\)
    • Variance (\(\sigma^2\)): \(\sigma^2=\lambda\)

Traffic Monitoring

  • Scenario: Observing traffic accidents at a busy intersection where the average is 3 accidents per month.
  • Question: What is the probability of observing exactly 5 accidents in one month?
Solution: Using \(\lambda=3\), \[p(5)=\frac{3^5 e^{-3}}{5!}\approx0.1008\]

Quality Control in Manufacturing

  • Scenario: Inspecting new automobiles for surface defects, with an average of 2 defects per vehicle.
  • Question: What is the probability of finding no defects on a newly inspected vehicle?
Solution: Using \(\lambda=2\), \[p(0)=\frac{2^0 e^{-2}}{0!}=e^{-2}\approx0.1353\]

Poisson Distribution Functions in R

The dpois function calculates the probability mass function of the Poisson distribution, useful for finding (P(X = x)).

  • Syntax: dpois(x, lambda)
  • Example: Calculate \(P(X = 3)\) for \(\lambda = 2.2\)
probability_exactly_3 <- dpois(3, lambda = 2.2)
probability_exactly_3
[1] 0.1966387

Poisson Distribution Functions in R

The ppois function computes the cumulative probability \(P(X \leq x)\) of the Poisson distribution.

  • Syntax: ppois(q, lambda)
  • Example: Calculate \(P(X \leq 3)\) for \(\lambda=2.2\)
probability_up_to_3 <- ppois(3, lambda = 2.2)
probability_up_to_3
[1] 0.8193524