Mathematical Statistics

The Poisson Probability Distribution

Ruslan Muslumov, PhD

ADA University, School of Business

Information Communication Technologies Agency, Statistics Unit

2025-11-05

Overview


Mathematical Statistics

The Poisson Probability Distribution


Ruslan Muslumov, PhD

ADA University

Overview

📚 Today’s Journey

  • The Poisson Probability Distribution
  • Derivation from Binomial Distribution
  • Properties and Validity Proof
  • Poisson Approximation to Binomial
  • Mean and Variance
  • Real-World Applications

🎯 Learning Objectives

  • ✅ Understand the Poisson distribution definition
  • 🔢 Master the probability mass function
  • 📊 Apply Poisson approximation techniques
  • 🧮 Solve practical problems
  • 📈 Analyze industrial and ecological phenomena

Think-Pair-Share: Random Events

🤔 Think (1-2 minutes):

Consider events that occur randomly over time or space. Can you think of three examples from your daily life where events happen with some average rate but at unpredictable times?

👥 Pair (2-3 minutes):

Share your examples with a partner. Discuss:

  • What makes these events “random”?
  • Can you estimate the average rate of occurrence?
  • Are the events independent of each other?

🗣️ Share:

Let’s hear a few examples from the class and discuss how they relate to today’s topic.

The Poisson Probability Distribution

Definition and Motivation

Definition

The Poisson probability distribution is a discrete probability distribution that describes the number of events that occur within a fixed interval of time or space, given a known average rate of occurrence. It is named after the French mathematician Siméon Denis Poisson.

Description

Split up the time period into (n) subintervals so that:

  • P(one event happens in a subinterval) = (p),
  • P(no event happens in a subinterval) = (1-p),
  • P(more than one event happens in a subinterval) = 0

Remark

Then the total number of events in the given time interval is just the total number of subintervals that contain one event.

Derivation from Binomial Distribution

Letting (= np) and taking the limit of the binomial probability

[ p(y) = p^y(1 - p)^{n-y} ]

as (n ), we have

[ \[\begin{align} \lim_{n \to \infty} \binom{n}{y} p^y(1 - p)^{n-y} &= \lim_{n \to \infty} \frac{n(n - 1) \cdots (n - y + 1)}{y!} \left( \frac{\lambda}{n} \right)^y \left(1 - \frac{\lambda}{n}\right)^{n-y} \end{align}\] ]

Derivation (continued)

[ \[\begin{align} &= \lim_{n \to \infty} \frac{\lambda^y}{y!} \left(1 - \frac{\lambda}{n}\right)^{n} \frac{n(n - 1) \cdots (n - y + 1)}{n^y} \left(1 - \frac{\lambda}{n}\right)^{-y} \end{align}\] ]

[ \[\begin{align} &= \frac{\lambda^y}{y!} \lim_{n \to \infty} \left(1 - \frac{\lambda}{n}\right)^{n} \cdot \lim_{n \to \infty} \left(1 - \frac{\lambda}{n}\right)^{-y} \cdot \lim_{n\to \infty}\left((1-\frac{1}{n}) \cdots (1 - \frac{y-1}{n})\right) \end{align}\] ]

[ = ]

Formal Definition

Definition

A random variable (Y) is said to have a Poisson distribution if and only if

[ p(y) = e^{-}, y = 0,1,2,,> 0 ]

Example

Show that the probabilities assigned by the Poisson probability distribution satisfy the requirements that (0 p(y) ) for all (y) and

[ _{y}p(y) = 1 ]

Proof of Validity

Proof

Because (> 0), it is obvious that (p(y) > 0) for (y = 0, 1, 2, ), and that (p(y) = 0) otherwise.

Further,

[ {y=0}^{} p(y) = {y=0}^{} e^{-} = e^{-} _{y=0}^{} = e^{-} e^{} = 1 ]

because the infinite sum (_{y=0}^{} ) is a series expansion of (e^).

⚡ Quick Check

Question

Before we move to applications, can you identify:

  1. What parameter characterizes the Poisson distribution?
  2. What does this parameter represent?
  3. What values can (Y) take in a Poisson distribution?

Answers

  1. The parameter () (lambda)
  2. It represents the average rate of events in the given interval
  3. (Y) can be (0, 1, 2, 3, ) (any non-negative integer)

Applications and Examples

Example 1: Police Patrol

Example

Suppose that a random system of police patrol is devised so that a patrol officer may visit a given beat location (Y = 0, 1, 2, 3, ) times per half-hour period, with each location being visited an average of once per time period. Assume that (Y) possesses, approximately, a Poisson probability distribution.

Calculate the probability that the patrol officer will:

  • Miss a given location during a half-hour period
  • Visit once
  • Visit twice
  • Visit at least once

Solution: Police Patrol (Part 1)

Solution

For this example, the time period is a half-hour, and the mean number of visits per half-hour interval is (= 1). Then

[ p(y) = , y = 0, 1, 2, ]

The event that a given location is missed in a half-hour period corresponds to (Y = 0), and

[ P(Y = 0) = p(0) = = e^{-1} ]

Solution: Police Patrol (Part 2)

Solution (continued)

Similarly,

[ p(1) = = e^{-1} ]

and

[ p(2) = = ]

The probability that the location is visited at least once is the event (P(Y )). Then

[ P(Y ) = _{y=1}^{} p(y) = 1 - p(0) = 1 - e^{-1} ]

Example 2: Tree Seedlings

Example

A certain type of tree has seedlings randomly dispersed in a large area, with the mean density of seedlings being approximately five per square yard. If a forester randomly locates ten 1-square-yard sampling regions in the area, find the probability that none of the regions will contain seedlings.

Solution: Tree Seedlings

Solution

(Y), the number of seedlings per square yard, can be modeled as a Poisson random variable with (= 5). Thus,

[ P(Y = 0) = p(0) = = e^{-5} ]

The probability that (Y = 0) on ten independently selected regions is

[ (e{-5}){10} ]

because the probability of the intersection of independent events is equal to the product of the respective probabilities. The resulting probability is extremely small.

🌍 Real-World Connection

Applications in Nature and Society

The Poisson distribution appears in many contexts:

  • Ecology: Distribution of organisms in space
  • Quality Control: Number of defects in manufacturing
  • Telecommunications: Number of calls arriving at a call center
  • Healthcare: Number of patients arriving at an emergency room
  • Physics: Radioactive decay events
  • Traffic Engineering: Number of vehicles passing a point

Poisson Approximation to Binomial

When to Use Poisson Approximation

Approximation Conditions

The Poisson distribution can be used to approximate the binomial distribution when:

  • (n) is large (typically (n ))
  • (p) is small (typically (p ))
  • (= np) is moderate (typically ())

Why It Works

When these conditions hold, the binomial distribution with parameters (n) and (p) is approximately Poisson with parameter (= np).

Example 3: Binomial Approximation

Example

Suppose that (Y) possesses a binomial distribution with (n = 20) and (p = 0.1). Find the exact value of (P(Y )) using the table of binomial probabilities, Table: “Binomial Distribution Histograms”. Use the following table, to approximate this probability, using a corresponding probability given by the Poisson distribution. Compare the exact and approximate values for (P(Y )).

Reference Tool

Poisson Distribution Applet

Solution: Approximation Comparison

Solution

Exact Binomial Calculation:

According to Table, the exact value of (P(Y ) = 0.867).

Poisson Approximation:

If (W) is a Poisson-distributed random variable with (= np = 20 = 2), previous discussions indicate that (P(Y )) is approximately equal to (P(W )). Table 3, Appendix 3, gives

[ P(W ) = 0.857 ]

Comparison: The Poisson approximation is quite good, yielding a value that differs from the exact value by only 0.01 (about 1% relative error).

Poisson Probability Table

Poisson Probabilities Table

Mean and Variance

Theorem: Expected Value and Variance

Theorem

If (Y) is a random variable possessing a Poisson distribution with parameter (), then

[ = E(Y) = ^2 = V(Y) = ]

Key Insight

In the Poisson distribution, the mean equals the variance. This is a unique property that distinguishes it from other distributions!

  • Standard deviation: (= )
  • This relationship simplifies many calculations

Example 4: Industrial Accidents

Example

Industrial accidents occur according to a Poisson process with an average of three accidents per month. During the last two months, ten accidents occurred.

Questions:

  • Does this number seem highly improbable if the mean number of accidents per month, (), is still equal to 3?
  • Does it indicate an increase in the mean number of accidents per month?

Solution: Industrial Accidents

Solution

The number of accidents in two months, (Y), has a Poisson probability distribution with mean (= 2 = 6). The probability that (Y) is as large as 10 is

[ P(Y ) = _{y=10}^{} ]

By Poisson Distribution Applet or from the theorem:

[ = = 6, ^2 = = 6, = = 2.45 ]

Solution: Statistical Interpretation

Solution (continued)

The empirical rule tells us that we should expect (Y) to take values in the interval () with a high probability.

Notice that:

[ + 2= 6 + (2)(2.45) = 10.90 ]

Interpretation: The observed number of accidents, (Y = 10), does not lie more than (2) from (), but it is close to the boundary. Thus, the observed result is not highly improbable, but it may be sufficiently improbable to warrant an investigation.

Think-Pair-Share: Decision Making

🤔 Think (2 minutes):

You are the safety manager. Based on the analysis showing 10 accidents in 2 months (close to the (+ 2) boundary), would you:

  • Conclude this is just random variation?
  • Investigate potential safety issues?
  • Set a new monitoring threshold?

👥 Pair (3 minutes):

Discuss with your partner:

  • What additional information would you want?
  • At what threshold would you definitely investigate?
  • How would you communicate this to management?

Summary and Key Takeaways

Key Takeaways

📝 Summary

Theoretical Foundation:

  1. PMF: (p(y) = )
  2. Parameter: (> 0) (rate parameter)
  3. Support: (y = 0, 1, 2, )
  4. Derived from: Binomial limit as (n )

Practical Properties:

  1. Mean = Variance = ()
  2. Approximates binomial when (n) large, (p) small
  3. Models rare events over time/space
  4. Independent occurrences assumption

Applications Review

🌟 We Explored

  1. Police patrol patterns - Rare event modeling
  2. Ecological distributions - Spatial patterns in nature
  3. Binomial approximation - Computational efficiency
  4. Industrial accidents - Statistical decision-making

📚 Next Steps

  • Practice with additional examples
  • Explore the relationship with the exponential distribution
  • Study applications in your field of interest
  • Review the empirical rule for statistical interpretation

Thank You!

📧 Questions?

Feel free to reach out during office hours or via email.

Course Resources:

📖 References

  • Wackerly, D., Mendenhall, W., & Scheaffer, R. Mathematical Statistics with Applications
  • Online resources: Poisson Distribution Calculator and Visualization Tools