Discrete Probability Distributions: Geometric and Poisson Distributions

Discrete Probability Distributions

Overview

Discrete probability distributions are fundamental in statistics for modeling situations where outcomes are countable and distinct. This section focuses on two important types: the Geometric Distribution and the Poisson Distribution, both of which are widely used in real-world applications.

Geometric Distribution

Definition and Properties

The geometric distribution is a discrete probability distribution that models the number of trials needed to achieve the first success in a sequence of independent trials, each with the same probability of success.

• Repeated Trials: The experiment consists of repeated trials until the first success occurs.

• Independence: Each trial is independent of the others.

• Constant Probability: The probability of success, denoted as p, remains constant for each trial.

• Random Variable: The random variable x represents the trial number on which the first success occurs.

The probability that the first success occurs on trial x is given by:

$P(x) = q^{x-1} \cdot p$ where q = 1 - p is the probability of failure.

Example: Business Failure Rate

Suppose the failure rate of businesses after five years is 50%. Four businesses that started five years ago are selected at random. Find the probability that the fourth business selected is the first one to have failed.

• Given: p = 0.50, q = 0.50, x = 4

• Calculation: $P(4) = 0.50^{3} \cdot 0.50 = 0.0625$

• Result: The probability is approximately 0.063.

Application: The geometric distribution is useful in quality control, reliability testing, and scenarios where the timing of the first occurrence is important.

Poisson Distribution

Definition and Properties

The Poisson distribution is a discrete probability distribution used to model the number of times an event occurs in a fixed interval of time, area, or volume.

• Counting Events: The experiment involves counting occurrences of an event within a specified interval.

• Constant Rate: The probability of the event occurring is the same for each interval.

• Independence: The number of occurrences in one interval is independent of other intervals.

• Mean: The mean number of occurrences is denoted by λ (lambda).

The probability of exactly x occurrences in an interval is:

$P(x) = \frac{\lambda^{x} e^{-\lambda}}{x!}$

Example: Intersection Accidents

The mean number of accidents per month at a certain intersection is three. What is the probability that in any given month four accidents will occur?

• Given: λ = 3, x = 4

• Calculation: $P(4) = \frac{3^{4} e^{-3}}{4!}$

• Result: Substitute values to compute the probability.

Application: The Poisson distribution is commonly used in fields such as biology, traffic analysis, and telecommunications to model rare events.

Example: Using Poisson Tables

A population count shows that the average number of rabbits per acre living in a field is 3.6. Use a table to find the probability that seven rabbits are found on any given acre.

• Given: λ = 3.6, x = 7

• Table Lookup: Using a Poisson probability table, the probability is found to be 0.0425.

• Technology Check: Using Excel, the probability is 0.042484, which rounds to 0.0425.

Application: Poisson tables and technology tools are useful for quickly finding probabilities without manual calculation.

Comparison of Geometric and Poisson Distributions

Key Differences

Distribution	Scenario	Parameter(s)	Probability Formula
Geometric	First success in repeated trials	p (probability of success)	$P(x) = q^{x-1} \cdot p$
Poisson	Number of events in interval	λ (mean number of events)	$P(x) = \frac{\lambda^{x} e^{-\lambda}}{x!}$

Visual Reference

The cover image below represents the textbook "Elementary Statistics" by Ron Larson, which is the source of these study notes.

Additional info: These notes expand on the brief points in the original material, providing definitions, formulas, and examples for clarity and completeness.