Calculate expected values, variances, and probabilities for any Poisson process.

An exam question rarely asks you to "define" a Poisson process. It asks for a number: a probability, a mean, or a variance. The mechanics below are the calculation engine.

The core probability formula. For a homogeneous Poisson process with rate $\lambda$, the count over an interval of length $t$ is Poisson with parameter $\lambda t$.

$$P(N(t)=k) = \frac{e^{-\lambda t}(\lambda t)^{k}}{k!}, \qquad k=0,1,2,\ldots$$

Plug in $\lambda t$ as a single number. If $\lambda=3$ per hour and $t=4$ hours, the parameter is 12. Both the mean and the variance equal 12.

KEY: The Poisson parameter is always $\lambda \times t$. The single most common error is using $\lambda$ when $t \neq 1$.

Cumulative probabilities. For "at least one" or "at most two" questions, sum or complement.

• $P(N(t)\ge 1) = 1 - e^{-\lambda t}$.
• $P(N(t)\le 2) = e^{-\lambda t}\bigl(1 + \lambda t + (\lambda t)^2/2\bigr)$.
• $P(N(t)\ge 3) = 1 - P(N(t)\le 2)$.

Inter-arrival and waiting times. Gaps $T_i$ between consecutive events are iid Exponential($\lambda$).

Common mistakes

• Forgetting to multiply $\lambda$ by $t$. Writing $P(N(t)=k) = e^{-\lambda}\lambda^k/k!$ when $t \ne 1$ drops the time factor. The Poisson parameter is always $\lambda t$ (or $\int \lambda(s)\,ds$ for NHPP).
• Using $\text{Var}(X)$ instead of $E[X^2]$ in compound Poisson variance. The correct formula is $\text{Var}(S) = \lambda t\,E[X^2]$. Always rebuild $E[X^2] = \text{Var}(X) + (E[X])^2$ first.
• Evaluating NHPP intensity at one point. For $\lambda(s) = 3 + 0.5s$ over $[4,10]$, the answer is not $\lambda(4) \times 6$ or $\lambda(10) \times 6$. Integrate.

Bottom line

• $N(t) \sim \text{Poisson}(\lambda t)$ so $E[N(t)] = \text{Var}(N(t)) = \lambda t$; probabilities use $P(N(t)=k) = e^{-\lambda t}(\lambda t)^k / k!$.
• Inter-arrival times $T_i \sim \text{Exponential}(\lambda)$, iid, with $E[T_i]=1/\lambda$ and $\text{Var}(T_i)=1/\lambda^2$; waiting time $S_n \sim \text{Gamma}(n,\lambda)$ with mean $n/\lambda$.
• Superpose by adding rates (only when the constituent processes are independent). Thin by classification probability $p$: independent Poisson sub-processes at rates $\lambda p$ and $\lambda(1-p)$.
• Memorylessness applies to gaps only, never to counts; observing past events does not change expected future arrivals.

Exam shortcut

When a problem fixes the total count $N(t)=n$ and asks about timing or types, switch frames: arrival times are uniform on $[0,t]$ and type labels are Binomial($n,p$). This collapses many Poisson questions to elementary distributions. For "at least one" probabilities, always complement to $1 - e^{-\lambda t}$ rather than summing the tail. The complement is one exponential evaluation; the tail is an infinite sum.

The full lesson (about 2,519 words, 17 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A2

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