Poisson Distribution: Concepts, Applications, and Calculations

Poisson Distribution

Introduction to Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided these events happen with a known constant mean rate and independently of the time since the last event.

• Poisson Variable: X = number of occurrences in a given interval (time, area, volume, etc.).

• Parameter: λ (lambda) = average rate of occurrence per interval.

• Key Properties:

  • Events occur independently.

  • The probability of more than one event in an infinitesimally small interval is negligible.

  • The average rate (λ) is constant.

Example: If birds land on a feeder at an average rate of 1.8 birds per hour, the number of birds landing in one hour can be modeled by a Poisson distribution with λ = 1.8.

Comparing Binomial and Poisson Experiments

Binomial Experiment	Poisson Experiment
Only 2 outcomes (success/failure) Fixed # of trials (n) Independent trials Equal probability of success (p)	Fixed time interval or region Independent events Occurrence = event happening (e.g., bird landing) Equal mean rate (λ) X = # of events in interval

Example: A bakery wants to predict how many customers will enter in 15 minutes. If the average is 2 customers per 15 minutes, the number of arrivals can be modeled by a Poisson distribution with λ = 2.

Poisson Probability Formula

The probability of observing exactly k events in a fixed interval is given by:

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

• Where:

  • k = number of occurrences (0, 1, 2, ...)

  • λ = mean number of occurrences in the interval

  • e ≈ 2.71828 (Euler's number)

Example: If the average number of birds landing in 1 hour is 1.8, the probability that exactly 3 birds land is: $P(X = 3) = \frac{e^{-1.8} (1.8)^3}{3!}$

Applications of the Poisson Distribution

• Counting the number of arrivals at a service point (e.g., customers at a bakery, cars at an intersection).

• Counting the number of defects in a length of material (e.g., flaws per meter of fabric).

• Modeling rare events in large populations (e.g., mutations in a population of mice).

Example: If a textile inspector finds an average of 0.5 defects per meter, the probability of finding 0 defects in a meter is: $P(X = 0) = \frac{e^{-0.5} (0.5)^0}{0!}$

Finding Probabilities Using the Poisson Distribution

• To find the probability of k or fewer events: sum the probabilities for all values from 0 to k.

• For cumulative probabilities, use the Poisson cumulative distribution function (CDF).

Example: Probability that 4 or fewer customers enter the bakery in 15 minutes (λ = 2): $P(X \leq 4) = \sum_{k=0}^{4} \frac{e^{-2} 2^k}{k!}$

Using the Poisson Distribution to Approximate Binomial Probabilities

When the number of trials n is large and the probability of success p is small, the Poisson distribution can approximate the binomial distribution:

• Requirements: $n \geq 100$, $np \leq 10$

• Set $\lambda = np$

Example: In a school raffle, each ticket has a 1/500 chance of winning. If 600 students each buy one ticket, the probability that 2 students win is: $\lambda = 600 \times \frac{1}{500} = 1.2$ $P(X = 2) = \frac{e^{-1.2} (1.2)^2}{2!}$

Finding Poisson Probabilities Using a TI-84 Calculator

• For exact probabilities: use poissonpdf(λ, k)

• For cumulative probabilities: use poissoncdf(λ, k)

Example: To find the probability that exactly 15 cars pass through an intersection in 10 minutes (λ = 17.4): poissonpdf(17.4, 15)

Example: To find the probability that at most 15 cars pass through: poissoncdf(17.4, 15)

Summary Table: Key Features of Poisson vs. Binomial Distributions

Feature	Binomial	Poisson
Type of Experiment	Fixed number of trials	Fixed interval (time/space)
Parameter(s)	n (trials), p (probability)	λ (mean rate)
Random Variable	Number of successes in n trials	Number of events in interval
Independence	Trials independent	Events independent
Application	Finite, repeated trials	Rare events, continuous monitoring

Additional info:

• The Poisson distribution is especially useful for modeling rare events in large populations or over continuous intervals.

• Standard deviation of a Poisson distribution is $\sqrt{\lambda}$.