Poisson process

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{title2=$W_n=T_n-T_{n-1}\sim\operatorname{Exp}(\lambda)$}

= Waiting times of a Poisson process
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{synonym}

Let $T_n=\inf\{t\geq0:N_t\geq n\}$ be the $n$th arrival time, and set $T_0=0$. In a <Homogeneous Poisson process> of rate $\lambda>0$, the gaps $W_n=T_n-T_{n-1}$ are <independent and identically distributed random variables> with an <exponential distribution>:
$$
\mathbb P(W_n>w)=e^{-\lambda w},\qquad f_{W_n}(w)=\lambda e^{-\lambda w},\qquad w\geq0;
\qquad \mathbb E[W_n]=\frac1\lambda,\quad \operatorname{Var}(W_n)=\frac1{\lambda^2}.
$$
Here $\operatorname{Exp}(\lambda)$ uses the rate convention. Under the scale convention $\theta=1/\lambda$, the same <probability density function> is $\theta^{-1}e^{-w/\theta}$; a simulation function accepting a scale must receive $1/\lambda$. Conversely, summing <independent> rate-$\lambda$ <exponential random variables> to form $T_n$ and counting those sums constructs a <Homogeneous Poisson process>.<Lectures on the Poisson Process by Günter Last and others>[Theorem 7.2, printed p. 59: interval theorem characterizing independent exponential gaps]{ref}

The first-gap formula follows immediately from $\{W_1>w\}=\{N_w=0\}$ and the <Poisson distribution>. Independence of all successive gaps is the additional interval-theorem assertion; deterministic-time <independent increments> alone should not simply be substituted at random arrival times without justification. The <memorylessness of the exponential distribution> gives
$$
\mathbb P(W_n>a+w\mid W_n>a)=e^{-\lambda w},\qquad a,w\geq0.
$$
At the toy rate of two per minute, the <mean> gap is half a minute, the <median> is $(\log2)/2\approx0.346574$ minutes, and the <probability> of waiting longer than half a minute is $e^{-1}\approx0.367879$. Here $\log$ is the <natural logarithm>. Already having waited a minute without an arrival leaves this same <conditional probability> of waiting another half minute. The <exponential distribution> describes a model with no aging of the remaining wait; a scheduled arrival with a fixed gap has no such property.


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 Incoming links: Poisson interarrival times