Poisson Distribution: Concepts, Formulas, and Applications

Section 4.5: Poisson Distribution

Introduction to the Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that these events happen independently and at a constant average rate. It is widely used in statistics to describe random events such as arrivals, failures, or occurrences.

• Poisson random variable (rv): A variable that counts the number of events in a given interval.

• Notation: $X \sim \text{Poi}(\mu)$, where $\mu$ is the average rate (mean) of occurrence.

• Typical applications: Number of phone calls per hour, number of accidents per month, number of defects per unit, etc.

Probability Mass Function (PMF) of the Poisson Distribution

The probability that a Poisson random variable $X$ takes the value $x$ is given by:

• Formula: $P(X = x) = \frac{e^{-\mu} \mu^x}{x!}$ where:

  • $\mu$ = mean number of occurrences in the interval

  • $x$ = number of occurrences ($x = 0, 1, 2, \ldots$)

  • $e$ = base of the natural logarithm ($\approx 2.718$)

• Expected value (mean): $E(X) = \mu$

• Variance: $\text{Var}(X) = \mu$

Worked Examples

Example 1: Plates Broken at a Restaurant

Suppose at a particular restaurant, an average of 4 plates are broken each week. Let $X$ be the number of plates broken each week, so $X \sim \text{Poi}(4)$.

• Probability that 2 plates are broken next week: $P(X = 2) = \frac{e^{-4} 4^2}{2!} = 0.1465$

Example 2: At Most 1 Plate Broken

• Probability that at most 1 plate is broken next week: $P(X \leq 1) = P(X = 0) + P(X = 1)$ $= \frac{e^{-4} 4^0}{0!} + \frac{e^{-4} 4^1}{1!}$ $= 0.0916$

Example 3: More Than 3 Plates Broken

• Probability that more than 3 plates are broken next week: $P(X > 3) = 1 - P(X \leq 3)$ $= 1 - \left( \frac{e^{-4} 4^0}{0!} + \frac{e^{-4} 4^1}{1!} + \frac{e^{-4} 4^2}{2!} + \frac{e^{-4} 4^3}{3!} \right)$ $= 0.76212$

Key Properties of the Poisson Distribution

• Discrete distribution: Only integer values ($x = 0, 1, 2, ...$) are possible.

• Events are independent: The occurrence of one event does not affect the probability of another.

• Constant mean rate: The average rate $\mu$ is constant over the interval.

• Variance equals mean: $\text{Var}(X) = E(X) = \mu$

Visual Representation

The Poisson distribution is often visualized as a bar graph showing the probability of each possible value of $x$. For example, in the case of plates broken, the probabilities for $x = 0, 1, 2, 3, ...$ can be plotted to show the likelihood of each outcome.

Summary Table: Poisson Distribution Formulas and Properties

Property	Formula
Probability Mass Function (PMF)	$P(X = x) = \frac{e^{-\mu} \mu^x}{x!}$
Mean	$E(X) = \mu$
Variance	$\text{Var}(X) = \mu$
Support	$x = 0, 1, 2, \ldots$

Applications of the Poisson Distribution

• Modeling the number of arrivals (e.g., customers, phone calls) in a fixed time period

• Counting the number of defects in a batch of products

• Estimating the frequency of rare events (e.g., accidents, system failures)

Additional info:

• The Poisson distribution is a limiting case of the binomial distribution when the number of trials is large and the probability of success is small, such that the expected number of successes ($np$) remains constant.

• Cumulative probabilities (e.g., $P(X \leq k)$) can be calculated by summing the PMF for all $x$ up to $k$.