FCSP - staging and grading

The Poisson Distribution

• It is a discrete probability distribution.
• It expresses the probability of a number of events happening in a fixed time or space interval.
• These events occur at a known constant mean rate.
• Each event happens independently after the last.

Definition

• A discrete random variable $X$ has a Poisson distribution with parameter $\lambda > 0$.
• The probability mass function (PMF) is: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$, $k = 0, 1, 2,...$
• $e$ is Euler's number ($e \approx 2.71828$).
• $k$ is the number of event occurrences.
• $k!$ is the factorial of $k$.
• $\lambda$ is a positive real number.
• $\lambda$ equals the expected number of occurrences during the interval.

Mean and Variance

• If $X \sim Poisson(\lambda)$, then
• $E[X] = \lambda$
• $Var(X) = \lambda$

Example

• Average goals in a soccer match = 2.
• The probability of 5 goals being scored in a match is:
• $P(X = 5) = \frac{e^{-2} 2^5}{5!} = 0.0361$

Poisson Sums

• Let $X_1, X_2,..., X_n$ be independent Poisson random variables.
• The means are $\lambda_1, \lambda_2,..., \lambda_n$, respectively.
• $X_1 + X_2 +... + X_n$ is a Poisson random variable.
• The mean is $\lambda_1 + \lambda_2 +... + \lambda_n$.

Example

• The number of emails arriving to the first account follows a Poisson distribution with a rate of 3 emails per hour.
• The number of emails arriving to the second account follows a Poisson distribution with a rate of 5 emails per hour.
• $X \sim Poisson(3)$ and $Y \sim Poisson(5)$.
• $X + Y \sim Poisson(3 + 5) = Poisson(8)$
• The probability that you will receive a total of 10 emails in one hour is:
• $P(X + Y = 10) = \frac{e^{-8} 8^{10}}{10!} = 0.09926$