given a mean number of events that happen within unit time (𝜆):

• the number of events occurring within that unit time has Poisson Distribution
• the time between events has Exponential Distribution
• the total time of 𝛼 events has Gamma Distribution

Probability Mass Function

𝐏(𝑋=𝑥) = 𝑒^−𝜆(𝜆^𝑥/𝑥!) for 𝑥 = 0, 1, 2, …

where:

• 𝜆 = frequency, mean number of events to happen in unit time
• 𝑒 = 2.7183… see number e (Euler’s number)
• 𝑥 = the number of “events” in question to happen in unit time

see: Deriving Poisson Distribution from Binomial Distribution

Expectation

𝐄[𝑋] = 𝜆

proof

• 𝐄[𝑋] = 𝛴_{0≤𝑥≤∞}[𝑥𝑓(𝑥)]
• 𝐄[𝑋] = 𝛴_{0≤𝑥≤∞}[𝑥𝑒^-𝜆𝜆^𝑥/𝑥!]
• 𝐄[𝑋] = 𝑒^-𝜆𝛴_{0≤𝑥≤∞}[𝑥𝜆^𝑥/𝑥!]
• 𝐄[𝑋] = 𝑒^-𝜆𝛴_{0≤𝑥≤∞}[𝜆^𝑥-1/(𝑥-1)!]𝜆
• 𝐄[𝑋] = 𝑒^-𝜆𝑒^𝜆𝜆 # via e taylor series
• 𝐄[𝑋] = 𝜆

Variance

𝑉𝑎𝑟(𝑋) = 𝜆

proof

Cumulative Distribution Function

𝐶𝐷𝐹(𝑋≤𝑥) = 𝛤(⌊𝑥+1⌋, 𝜆) / ⌊𝑥⌋!

where:

• 𝛤 - upper incomplete gamma function

Moment Generating Function$M_X(t)=e^{𝜆(e^t-1)}$

Example

Suppose you knew that the mean number of calls to a fire station on a weekday is 8. What is the probability that on a given weekday there would be 11 calls? This problem can be solved using the following formula based on the Poisson distribution

𝜆 = 8, therefore PMF is

• 𝐏(𝑋=𝑥) = 𝑒⁻⁸(8^𝑥/𝑥!)

what’s the probability that on a given weekday there would be 11 calls?

• 𝐏(𝑋=11) = 𝑒⁻⁸(8¹¹/11!)
• 𝐏(𝑋=11) = 0.072

Plot PMF given 𝜆=8, for 0≤𝑥≤12

Subpages

• Gamma Distribution vs Poisson Distribution
• Poisson vs Exponential
• Binomial Distribution vs Poisson Distribution