What is Poisson distribution?

The Poisson distribution is a discrete probability distribution that describes the likelihood of a certain number of events occurring within a fixed interval of time or space.

Probability mass function (PMF)

The probability mass function of the Poisson distribution is given by the formula:

Where:

$P(X = k)$ represents the probability of observing k events.

$\lambda$ is the average rate at which events occur within the given interval.

$e$ is Euler's number (approximately 2.71828).

$k!$ denotes the factorial of k.

Properties of Poisson distribution

Mean and variance: The mean and variance of a Poisson distribution are both equal to the rate parameter $λ$ , indicating that the average number of events and the spread of the distribution is determined by $λ$ .
Memoryless property: The Poisson distribution exhibits a memoryless property, meaning that the probability of an event occurring in a specific interval is independent of past events.
Independent events: In a Poisson distribution, the number of events occurring in non-overlapping intervals is independent of each other.

Real-world applications

Queuing theory: Poisson distribution is widely applied in queuing theory to analyze customers' arrival and service rates in various systems such as call centers and supermarkets.
Finance and stock market analysis: Poisson distribution can be applied to model the occurrence of extreme events or market shocks in finance and stock market analysis.
Natural disasters: Poisson distribution is applied in assessing the frequency of natural disasters, such as earthquakes or floods, helping understand their risk and informing disaster management strategies.

Example

The average number of cars passing through a specific intersection in a city is 5 per minute. What is the probability of observing exactly three cars passing through the intersection in a one-minute interval?

Solution

Here we can extract the following things from the question,

So, the probability of observing exactly three cars passing through the intersection in one minute, given an average rate of 5 cars per minute, is approximately 0.1403 or 14.03%.

Conclusion

The Poisson distribution is a powerful tool for modeling random events that occur at a fixed rate within a given time or space interval. Researchers, analysts, and decision-makers can make accurate predictions and optimize various processes in their respective domains by understanding and utilizing the Poisson distribution.