Poisson Distribution Explained

The Poisson Distribution

• The Poisson distribution is a discrete probability distribution.
• It expresses the probability of a number of events occurring in a fixed interval.
• These events occur with a constant mean rate.
• Events occur independently of the time since the last event.

Definition

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

Mean and Variance

• If $X \sim Poisson(\lambda)$, then:
• $E[X] = \lambda$ (the expected value of X is lambda)
• $Var(X) = \lambda$ (the variance of X is lambda)

Example

• Goals scored in a soccer match average 2.
• The probability of scoring 5 goals in a match can be modeled as follows:
• $P(X = 5) = \frac{e^{-2} 2^5}{5!} = 0.0361$

Poisson Sums

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

Example

• There are two email accounts.
• The first account receives emails following a Poisson distribution at a rate of 3 per hour.
• The second account receives emails following a Poisson distribution at a rate of 5 per hour.
• The probability of receiving a total of 10 emails in one hour can be found using Poisson Sums:
• Let X be the number of emails arriving to the first account, and Y be the number of emails arriving to the second account.
• $X \sim Poisson(3)$ and $Y \sim Poisson(5)$.
• $X + Y \sim Poisson(3 + 5) = Poisson(8)$
• $P(X + Y = 10) = \frac{e^{-8} 8^{10}}{10!} = 0.09926$