5.2 Poisson Distribution: Properties, Applications, and Approximations

Discrete Distributions: Poisson Distribution

Introduction to the Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed interval of time, area, or space. It is widely used in statistics, especially for rare events, and is applicable when events occur independently and at a constant average rate.

• Key applications: Number of phone calls per hour, accidents per day, product failures per month.

• Parameter: λ (lambda) represents the average number of occurrences per unit interval.

Definition and Assumptions

The Poisson distribution describes the probability of observing x events in a fixed interval, given:

• Events occur independently.

• The average rate (λ) is constant.

• No two events occur at exactly the same instant.

Notation: $X \sim \text{Poisson}(\lambda)$

Probability Mass Function (PMF)

The probability of observing x events is given by:

• For $x = 0, 1, 2, \ldots$:

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

• Otherwise: $p(x) = 0$

• Where $e \approx 2.718$ is the base of the natural logarithm.

Example: Probability Calculation

Scenario: A bottling machine breaks down on average once every two weeks. The number of breakdowns follows a Poisson distribution. What is the probability that the machine breaks down more than twice in the next two weeks?

• Let $X$ be the number of breakdowns per two weeks, $X \sim \text{Poisson}(1)$.

• Calculate $P(X > 2) = 1 - P(X \leq 2)$:

$P(X > 2) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))$ $= 1 - \left( \frac{1^0 e^{-1}}{0!} + \frac{1^1 e^{-1}}{1!} + \frac{1^2 e^{-1}}{2!} \right)$ $= 1 - (e^{-1} + e^{-1} + \frac{e^{-1}}{2})$ $= 1 - \frac{5e^{-1}}{2} \approx 1 - 0.9197 = 0.0803$

Poisson Approximation to the Binomial Distribution

When the number of trials is large and the probability of success is small, the binomial distribution can be approximated by the Poisson distribution.

• Binomial PMF: $p(x) = \binom{n}{x} p^x (1-p)^{n-x}$

• If $n$ is large, $p$ is small, and $np = \lambda$ is moderate, then:

$\lim_{n \to \infty} \binom{n}{x} p^x (1-p)^{n-x} = \frac{\lambda^x e^{-\lambda}}{x!}$

• Rule of thumb: $n \geq 100$ and $np \leq 10$

• This approximation is useful when direct binomial calculations are difficult.

Expected Value and Variance

For $X \sim \text{Poisson}(\lambda)$:

• Expected value: $E[X] = \lambda$

• Variance: $\text{Var}(X) = \lambda$

Derivation of Expected Value and Variance

The sum of the PMF over all possible values equals 1:

$\sum_{x=0}^{\infty} p(x) = \sum_{x=0}^{\infty} \frac{\lambda^x e^{-\lambda}}{x!} = 1$

Expected value:

$E[X] = \sum_{x=0}^{\infty} x p(x) = \lambda$

Variance:

$\text{Var}(X) = E[X^2] - (E[X])^2 = \lambda$

Conditions for Modeling with Poisson Distribution

Condition	Example
Count events in a fixed interval	Number of emails received per hour
Events occur independently	Email arrivals are generally unrelated
Constant average rate	Emails arrive at a steady rate
Rare in small intervals	Unlikely to receive multiple emails in one second

Other examples: Customer arrivals at a bank per hour, website clicks per minute.

Poisson vs Binomial: How to Decide

When to Use Poisson(λ)	When to Use Bin(n, p)
Counting events in a fixed interval (time, area, space)	Counting successes in a fixed number of trials
X can be any nonnegative integer: {0, 1, 2, ...}	X cannot exceed n: {0, 1, ..., n}
Example: Number of emails received in an hour	Example: Number of spam emails out of 10 received

Special case: Bin(n, p) with $n \geq 100$, $np \leq 10$ can be approximated by Poisson($np$).

Changing the Fixed Interval

When the unit of the interval changes, the average rate must be adjusted accordingly.

• If $X$ is the number of emails per hour, $X \sim \text{Poisson}(0.5)$

• For emails per day: $Y \sim \text{Poisson}(12)$, where $12 = 0.5 \times 24$

• For emails per minute: $Z \sim \text{Poisson}(0.0083)$, where $0.0083 \approx 0.5/60$

Additional Examples

• Bottling machine breakdowns: For four weeks, expected rate doubles: $Y \sim \text{Poisson}(2)$

• Probability of exactly 3 breakdowns: $P(Y = 3) = \frac{2^3 e^{-2}}{3!} \approx 0.1804$

• Expected number of breakdowns in four weeks: $E[Y] = 2$

Poisson Approximation Example

A publishing company typesets a novel with 300 pages and 1,500 letters per page. Probability of a typo per letter is 0.01. What is the probability of having 5 or fewer typos?

• Total letters: $n = 300 \times 1,500 = 450,000$

• $X \sim \text{Bin}(450,000, 0.01)$, $np = 4,500$

• Since $n > 100$ and $np$ is moderate, use Poisson approximation: $X \sim \text{Poisson}(4,500)$

• $P(X \leq 5) \approx 0.7029$

Historical Application

R. D. Clarke (1946) studied flying bomb hits in London during WWII. Over 537 bombs struck 144 km2 of London, divided into 576 squares (0.25 km2 each), giving an average of 0.93 bombs per square. The Poisson model ($X \sim \text{Poisson}(0.93)$) fit the observed data well.

References

• Telhammer, R. C. (2013). Mathematical Statistics for Economics and Business. Springer.

• Clarke, R. D. (1946). Application of the Poisson distribution. Journal of the Institute of Actuaries, 72(3), 481-481.

• Shaw, L. P., & Shaw, L. F. (2019). The flying bomb and the actuary. Significance, 16(5), 12-17.

Additional info:

• Midterm announcement and exam preparation tips were included in the original notes but are not directly relevant to Poisson distribution content.