Probability

Poisson Process

Arrivals over continuous time with rate $\lambda$

Arrivals

 

Cumulative count N(t)

Key properties

  • Interarrival times are exponential with rate $\lambda$, meaning they're memoryless.
  • $N(t) \sim \text{Poisson}(\lambda t)$. The count in any interval of length t follows a Poisson distribution.
  • $\mathbb{E}[N(t)] = \mathbb{V}(N(t)) = \lambda t$. Expectation and variance are equal.
  • Disjoint intervals are independent. What happens in [0,5] tells you nothing about [5,10].