Poisson Distribution

What is Poisson Distribution? Poisson Distribution Explained

The Poisson distribution is a discrete probability distribution that describes the number of events that occur within a fixed interval of time or space, given a known average rate of occurrence. It is often used to model rare or random events, where the events occur independently and at a constant average rate.

It is defined by a single parameter, denoted by λ (lambda), which represents the average number of events that occur in the given interval. The probability mass function (PMF) of the Poisson distribution is given by:

P(X=k) = (e^(-λ) * λ^k) / k!

Where:

P(X=k) is the probability that the number of events (X) is equal to k.
e is the base of the natural logarithm (approximately 2.71828).
λ is the average rate of events per interval.
k is the number of events (k = 0, 1, 2, …).

Some key properties and characteristics of this distribution are:

Mean and Variance: The mean (μ) and variance (σ^2) of a Poisson distribution are both equal to λ.

Memoryless Property: It exhibits the memoryless property, which means that the probability of an event occurring within a given time period is independent of the time that has already passed.

Rare Events: It is often used to model rare events that occur at a low average rate. As λ increases, the distribution becomes more concentrated around its mean, resembling a normal distribution.

Independent Events: The events modeled by this probability distribution are assumed to occur independently of each other. The occurrence of one event does not affect the probability of another event happening.

The Poisson distribution has various applications in different fields, including:

• Modeling the number of phone calls received at a call center within a specific time period.
• Analyzing the number of accidents occurring on a highway in a given hour.
• Estimating the number of email spam messages received in a day.
• Studying the number of defects in a manufacturing process within a specific time frame.
• Analyzing the occurrence of earthquakes in a particular region over a given period.

It is important to note that the Poisson distribution assumes that the events occur randomly and independently at a constant average rate. If the events are not truly independent or occur at irregular rates, other distributions such as the negative binomial distribution or the compound Poisson distribution may be more appropriate.