The Poisson distribution is a probability distribution that describes the likelihood of a given number of events occurring in a fixed time period or in a fixed area. It is often used to model the number of times an event occurs in a given period, such as the number of customers arriving at a store in a given hour or the number of defective items in a batch of products.

## Properties of Poisson Distribution:

The Poisson distribution is defined by a single parameter: the average number of events per time period or area. It has several important properties, including:

**Discrete values:**The Poisson distribution only takes on integer values, meaning that the number of events can only be a whole number.**Two possible outcomes:**The experiment results in outcomes that can be classified as successes or failures. In the context of the Poisson distribution, a success is an event or trial that meets the criteria of interest, while a failure is an event or trial that does not meet the criteria of interest. For example, in a problem involving the number of customer arrivals at a store, a success could be a customer arrival.**The average number of successes are known:**The average number of successes (μ) that occurs in a specified region is known. To use the Poisson distribution to model a problem, it is necessary to know the average rate of occurrence of the events or trials of interest.**Outcomes are random:**This means that the occurrence of one outcome does not influence the likelihood of another outcome of interest. In other words, the events or trials modelled using the Poisson distribution are independent of each other, and the outcome of one event or trial does not affect the outcome of another event or trial.**Infinite possible values:**The Poisson distribution can take on any positive integer value from 0 to infinity. The outcomes of interest are rare relative to the possible outcomes. In other words, the events or trials being modelled using the Poisson distribution are relatively infrequent compared to the total number of possible events or trials.**Mean and variance:**The Poisson distribution has a mean equal to the average number of events per time period or area and a variance equal to the mean. This means that the mean and variance of a Poisson distribution are equal.

An example of a problem that satisfies these properties of the Poisson distribution is the number of road accidents occurring in a given area over a certain period. In this problem, the outcomes can be classified as successes (road accidents) or failures (no road accidents), the average number of road accidents is known, the occurrence of one road accident does not influence the likelihood of another road accident, and road accidents are rare relative to the total number of possible outcomes (the total number of vehicle miles travelled in the area).

## Probability Mass Function (PMF) - Poisson Distribution

A Poisson distribution's probability mass function (PMF) gives the probability of a given number of events occurring in a fixed time period or area. The formula for the PMF is as follows:

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

Where:

\(P(x)\) is the probability of x events occurring in a fixed time period or area

\(\lambda\) is the average number of events per time period or area (x) is the number of events

\(!\) is the factorial symbol, which denotes the product of all positive integers less than or equal to the number. For example, \(4! = 4 * 3 * 2 * 1 = 24\).

The formula for the Poisson distribution uses the exponential function \(e^{-\lambda}\) and factorials to calculate the probability of a given number of events occurring in a fixed time period or area.

## Mean, Standard Deviation and Variance of Poisson Distribution

The mean, variance, and standard deviation are important statistical measures that describe the characteristics of a probability distribution. In the case of the Poisson distribution, the mean and variance are equal to the average number of events per time period or area \(\lambda\). This means that the Poisson distribution is completely determined by its mean value.

## Using Microsoft Excel:

One way to calculate the probability of a Poisson event is by using Microsoft Excel. Excel provides a number of functions that can be used to perform statistical calculations, including the POISSON.DIST function that calculates the probability of a specified number of events occurring in a fixed time period or area.

To use the POISSON.DIST(x, mean, cumulative) function, you need to provide the following input arguments:

- x: The number of events you want to calculate the probability for.
- mean: The average number of events per time period or area.
- cumulative: A logical value that specifies whether to return the probability of the specified number of events (FALSE) or the probability of the specified number of events or fewer (TRUE).

For example, in a new worksheet, enter the following formula into a cell:

=POISSON.DIST(7, 3.6, TRUE)

This formula calculates the probability of exactly 7 events occurring in a 10-minute time period with an average rate of 3.6 events per 10 minutes. The third argument, TRUE, indicates that the function should return the cumulative Poisson probability, which is the probability of the specified number of events or fewer occurring.

After entering the formula, press Enter to evaluate it, and the result 0.9692 will be displayed in the cell. This is the probability of 7 or fewer people arriving at the counter in 10 minutes, as calculated using the Poisson distribution.

If you want to find out the probability of exactly 7 customers, then you can use:

=POISSON.DIST(7, 3.6, FALSE)

This will give the output as 0.04248 (roughly 4%).

## Applications of Poisson Distribution

The Poisson distribution has many practical applications in fields such as engineering, economics, and biology. Some common examples include:

- Modelling the number of phone calls received by a call center in a given hour
- Predicting the number of customers who will arrive at a store in a given hour
- Modelling the number of defects in a batch of products
- Estimating the number of accidents that will occur on a stretch of road in a given year

## Conclusion

The Poisson distribution is a useful tool for modelling the number of events occurring in a fixed period or area. It is defined by a single parameter: the average number of events per time period or area, and it has a mean and variance equal to this parameter.