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:

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