The 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, assuming these events occur with a known constant mean rate and independently of the time since the last event.
NoteNamed after French mathematician Siméon Denis Poisson, this distribution plays a crucial role in probability theory and statistics, particularly in modeling rare events.
Mathematical Definition
The Poisson distribution is defined by its probability mass function:
$$P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$$
Where:
- $X$ is the random variable representing the number of events
- $k$ is the number of occurrences $(k = 0, 1, 2, ...)$
- $e$ is Euler's number (approximately 2.71828)
- $\lambda$ is the expected number of occurrences that occur during the given interval
Mean and Variance
One of the remarkable properties of the Poisson distribution is that its mean and variance are equal:
$$\text{Mean} = \text{Variance} = \lambda$$
HintThis property, known as equidispersion, is unique to the Poisson distribution and can be useful in identifying whether a dataset follows a Poisson distribution.
NoteThe equality of mean and variance in the Poisson distribution is a key characteristic that distinguishes it from other discrete distributions like the binomial distribution.
Conditions for Poisson Distribution
For a situation to be appropriately modeled by a Poisson distribution, two key conditions must be met:
- Independence of Events:
- Each event must occur independently of all other events.
- This means that the occurrence of one event does not affect the probability of another event occurring.
- Uniform Average Rate:
- Events must occur at a constant average rate within the period of interest.
- This rate should not change over time or space within the interval being considered.
Consider the number of customers arriving at a small coffee shop between 2 PM and 3 PM on weekdays. If, on average, 20 customers arrive during this hour, and their arrivals are independent of each other and occur at a roughly constant rate, this scenario could be modeled using a Poisson distribution with $\lambda = 20$.
Non-Overlapping Intervals in Poisson Distribution
An essential condition for the Poisson distribution is that events occurring in non-overlapping intervals are independent. This means that the number of events in one interval does not affect the number of events in another interval.
Example- If we model the arrival of buses at a station using a Poisson distribution, knowing that two buses arrived in the first hour should not change the probability of how many arrive in the next hour.
- Each time period is considered separately, and events occurring in one do not influence another.
Sum of Independent Poisson Distributions
An important property of the Poisson distribution is that the sum of two independent Poisson distributions is also a Poisson distribution.
