Lagmental Vicfred

Exponential Waiting Times Are Memoryless by Vicfred

tags: mathematics, stochastic processes, order statistics, information theory

The time to the next Poisson arrival has the unique continuous memoryless distribution. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

Continuous-time processes are described by finite-dimensional distributions plus path regularity. A Poisson process \(N_t\) has independent increments, while Brownian motion \(B_t\) has Gaussian increments.

$$ T\sim\operatorname{Exp}(\lambda) $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \mathbf P(T>s+t\mid T>s)=\mathbf P(T>t) $$

A small case in full

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \frac{e^{-\lambda(s+t)}}{e^{-\lambda s}}=e^{-\lambda t},\qquad f_T(t)=\lambda e^{-\lambda t}\mathbf1_{\{t\ge0\}} $$

The reusable statement

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad T\sim\operatorname{Exp}(\lambda),\\[5pt] \mathsf{C}\;&:\quad \mathbf P(T>s+t\mid T>s)=\mathbf P(T>t). \end{aligned} $$

A nearby false statement

Matching means and variances does not identify a distribution. Independence, increment laws, and sample-path properties are separate ingredients.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf P(T>s+t\mid T>s)=\mathbf P(T>t) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Fri 18 November 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.