Discrete Probability Distributions

Overview

This section explores two important types of discrete probability distributions: the Geometric Distribution and the Poisson Distribution. These distributions are used to model different types of random processes and are fundamental in statistical analysis of discrete events.

Geometric Distribution

Definition and Properties

• Geometric Distribution: A discrete probability distribution that models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials (trials with two possible outcomes: success or failure).

• Conditions:

  • Each trial is independent.

  • The probability of success, denoted as p, is constant for each trial.

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

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

Example: Using the Geometric Distribution

• Scenario: The success rate of businesses after five years is 52%. Four businesses are selected at random. Find the probability that the fourth business selected is the first one to have succeeded.

• Solution:

  • Let p = 0.52 (probability of success), q = 0.48 (probability of failure), and x = 4.

  • Probability that the first three businesses failed and the fourth succeeded:

  • So, the probability is approximately 0.058.

Poisson Distribution

Definition and Properties

• Poisson Distribution: A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time, area, or volume.

• Conditions:

  • The experiment counts the number of times an event occurs in a given interval.

  • The probability of the event is the same for each interval.

  • Occurrences in different intervals are independent.

The probability of exactly x occurrences in an interval is given by:

where μ is the mean number of occurrences in the interval, e is the base of the natural logarithm, and x! is the factorial of x.

Example: Using the Poisson Distribution

• Scenario: 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?

• Solution:

  • Let μ = 3 and x = 4.

  • Calculate the value using a calculator or statistical table.

Example: Finding a Poisson Probability Using a Table

• Scenario: The average number of rabbits per acre in a field is 3.6. Find the probability that seven rabbits are found on any given acre.

• Solution:

  • Let μ = 3.6 and x = 7.

  • Using a Poisson table or technology (e.g., Excel), the probability is approximately 0.0425.