Poisson Distribution

The Poisson distribution is a discrete v.a. \xi which measures the number of times an event occurs in a time or space interval and is denoted as:

\xi \approx P(\lambda)

Its probability function is:

P(\xi = k ) = e^{-\lambda} \cdot \frac{\lambda^k}{k!}, k \in \{0, \cdots, n\}

E(\xi) = \lambda

\sigma^2(\xi) = \lambda

\sigma(\xi) = +\sqrt{\lambda}

Properties

1. \xi = \xi_1 + \xi_2 \approx P(\lambda) with \lambda = \lambda_1 + \lambda_2 when \xi_1, \xi_2 are independent v.a.
2. \xi = \xi_1 + \cdots + \xi_r \approx P(\lambda) with \lambda = \lambda_1 + \cdots + \lambda_r when \xi_1, \cdots, \xi_r are independent v.a.

Approximation of the Binomial to the Poisson

It \xi \approx P(\lambda)\approx B(n, p)

If \exists \lim\limits_{n\to\infty, p\to 0}n\cdot p = \lambda \Rightarrow \lim\limits_{n\to\infty, p\to 0}P(\xi = k ) = e^{-\lambda} \cdot \frac{\lambda^k}{k!}

with k \in \{0, \cdots, n\}

That is, a Binomial, where bernoulli's number of tests is large (n tends to infinity) and the probability of success in each test is small (p tends to 0) is approximately one Poisson parameter \lambda=n\cdot p

It is considered a good approximation when n \geq 50 and p \leq 0.1

Calculation of a Poisson

\lambda k

\xi \approx P(\lambda)

E(\xi)
\sigma^2(\xi)
\sigma(\xi)