Waiting Times (Poisson)

Definition

Let

• $\{N(t),t\ge 0\}$ be a Poisson process of rate $\lambda$.
• $T_i$ are the inter-arrival times at $i$-th event

The Waiting Time $W_n$ is the time of occurrence of the $n$-th event:

$W_n={\sum }_{i=1}^n{T}_i$

Interpretation

The waiting time is the sum of $n$ independent exponential clocks. You need $n$ events to happen, and each takes exponential time — the total is gamma distributed.

Distribution

Then $W_n\sim \text{Gamma}(n,\lambda )$ with PDF: $f_{W_n}(t)=\lambda e^{-\lambda t}\frac{(\lambda t{)}^{n-1}}{(n-1)!},t\ge 0$

Key Relationship

$N(t)\ge n⟺W_n\le t$

The $n$-th event occurs by time $t$ if and only if at least $n$ events have occurred by time $t$.

Proof via Equivalence

$$\begin{array}{rl}{F}_{W_n}(t)& =P(W_n\le t)=P(N(t)\ge n)\\ & =\sum _{j=n}^{\infty }{e}^{-\lambda t}\frac{(\lambda t{)}^j}{j!}\end{array}$$

Differentiating with respect to $t$ yields the Gamma$(n,\lambda )$ PDF.

Related

• Poisson Process
• Inter-arrival Times
• Conditional Distribution of Arrival Times