5.3 Poisson Probability Distributions – Study Notes

5.3 Poisson Probability Distributions

Introduction to Poisson Probability Distributions

The Poisson probability distribution is a discrete probability distribution that describes the probability of a given number of events occurring in a fixed interval of time, distance, area, or volume. It is particularly useful for modeling rare events in large populations or over continuous intervals.

• Random Variable (x): Represents the number of occurrences of the event in the interval.

• Interval: Can be time, distance, area, volume, or similar units.

• Examples:

  • Number of automobile accidents in a day

  • Number of patients arriving at an emergency room in one hour

  • Number of internet users logging onto a website in a day

Poisson Probability Distribution Formula

The probability of observing exactly x occurrences in a given interval is calculated using the following formula:

$P(x) = \frac{\mu^x \cdot e^{-\mu}}{x!}$

• e: Mathematical constant, approximately 2.71828

• μ (mu): Mean number of occurrences in the interval

• x: Number of occurrences (0, 1, 2, ...)

Requirements for the Poisson Probability Distribution

For a situation to be modeled by a Poisson distribution, the following requirements must be met:

1. The random variable x is the number of occurrences of an event in some interval.

2. The occurrences must be random.

3. The occurrences must be independent of each other.

4. The occurrences must be uniformly distributed over the interval being used.

Parameters and Properties of the Poisson Distribution

• Parameter: The Poisson distribution is determined solely by the mean, μ.

• Possible Values: x can take values 0, 1, 2, ... with no upper limit.

• Mean: $\mu$

• Standard Deviation: $\sigma = \sqrt{\mu}$

Example: Atlantic Hurricanes

This example demonstrates the application of the Poisson distribution to real-world data.

• Given: 652 Atlantic hurricanes over 118 years.

• a. Find μ (mean number of hurricanes per year):

  • $\mu = \frac{652}{118} = 5.5$

• b. Find the probability of exactly 6 hurricanes in a year (P(6)):

  • Given x = 6, μ = 5.5, e = 2.71828

  • $P(6) = \frac{5.5^6 \cdot 2.71828^{-5.5}}{6!} = 0.157$

• c. Compare expected and actual results:

  • Expected number of years with 6 hurricanes: $118 \times 0.157 = 18.5$ years

  • Actual number of years with 6 hurricanes: 16 years

  • The Poisson model appears to fit the data reasonably well.

Poisson Distribution as an Approximation to the Binomial Distribution

The Poisson distribution can be used to approximate the binomial distribution when the number of trials (n) is large and the probability of success (p) is small. This is particularly useful when direct computation of binomial probabilities is difficult.

• Conditions for Approximation:

  • n ≥ 100

  • np ≤ 10

• Mean for Poisson Approximation:

  • $\mu = np$

Summary Table: Poisson Probability Distribution

Property	Description
Type	Discrete Probability Distribution
Parameter	Mean (μ)
Possible Values of x	0, 1, 2, ... (no upper limit)
Mean	μ
Standard Deviation	$\sqrt{\mu}$
Probability Formula	$P(x) = \frac{\mu^x \cdot e^{-\mu}}{x!}$
Approximation to Binomial	When n ≥ 100 and np ≤ 10, use μ = np