# Poisson Distribution - Mathematics
Data Science for Political and Social Phenomena
Author: Guillaume Redoulès

# Poisson Distribution

## Poisson Experiment

Poisson experiment is a statistical experiment that has the following properties: * The outcome of each trial is either success or failure. * The average number of successes (\(\lambda\)) that occurs in a specified region is known. * The probability that a success will occur is proportional to the size of the region. * The probability that a success will occur in an extremely small region is virtually zero.

## Poisson Distribution

A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution:

$$P(k,\lambda)=\frac{\lambda^ke^{-\lambda}}{k!}$$

where : * \(\lambda\) is the average number of successes that occur in a specified region. * \(k\) is the actual number of successes that occur in a specified region. * \(P(k,\lambda)\) is the Poisson probability, which is the probability of getting exactly \(k\) successes when the average number of successes is \(\lambda\).

## Example

The average number of goals in the soccer world cup is 2.5. The probability that 4 goals are scored is then:

$$p(\lambda=2.5,k=4)=\frac{2.5^4e^{-2.5}}{4!}=0.133$$

## Expectation for the Poisson distribution

Consider some Poisson random variable, \(X\). Let \(E[X]\) be the expectation of \(X\). Find the value of \(E[X^2]\).

Let \(Var(X)\) be the variance of \(X\). Recall that if a random variable has a Poisson distribution, then: * \(E[X]=\lambda\) * \(Var[X]=\lambda\)

Now, we'll use the following property of expectation and variance for any random variable, \(X\):

$$Var(X)=E[X^2]-(E[X])^2$$
$$E[X^2]=Var(X)+(E[X])^2$$

So, for any random variable having a Poisson distribution, the above result can be rewritten as:

$$E[X^2]=\lambda + \lambda^2$$

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