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MathidiaMathematics Academy
Stochastic processes
Lesson 38 of 75
Standard practice

Poisson processes

Objective. Model event times and independent increments.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

A Poisson process models random events occurring over time such that the number of events in any interval is Poisson-distributed and increments over disjoint intervals are independent. If events occur at rate λ per unit time, the number of events N(t) in an interval of length t satisfies N(t) ~ Poisson(λt), and the waiting time between consecutive events is exponentially distributed with rate λ. The key defining property is independent, stationary increments: the number of events in (s, s+t) depends only on t, not on s or on earlier history. Example: for a process with rate λ = 3 per hour, the probability of exactly 2 events in a 1-hour window is e^(−3)3²/2!.

Worked example

For a Poisson process with rate λ = 3 events per hour, find the probability of exactly 2 events occurring in a 1-hour interval.

3 … 2
Illustrated card showing 3 … 2
  1. 1The count N(1) in a 1-hour interval follows Poisson(λt) = Poisson(3).
  2. 2The Poisson probability mass function is P(N=k) = e^(−μ)μᵏ/k!, with μ = 3 and k = 2.
  3. 3Substitute: P(N=2) = e^(−3)(3²)/2! = e^(−3)(9)/2.
  4. 4Simplify: P(N=2) = 4.5e^(−3), which numerically is approximately 0.224.

Answer: P(N=2) = 4.5e⁻³ ≈ 0.224

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Common mistake. Students often confuse the inter-arrival time distribution with the count distribution; the number of events in a fixed interval is Poisson, but the gap between events is exponential, not Poisson, and conflating the two leads to wrong probability formulas.

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