Deriving the exponential distribution from the time between event in a Poisson distribution

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Notes

Let $X\sim$$\text{Poi}(\lambda)$ for some $\lambda\in\mathbb{R}_{>0}$

• Here supposed that $X$ models the number of events per unit time - although as with Poisson distribution - any continuum will do

Then:

• Suppose an event happens at $t\eq t_0$
  • Let $T$ be the random variable which is the time until the next event.
    • Let $d\in\mathbb{R}_{>0}$ be given, so we can investigate $\P{T>d}$
    • We are interested in $\mathbb{P}\big[\text{no events happening for time in }(t_0,t_0+d)\big]\eq\P{T>d}$
      • Let $X'\sim\text{Poi}(\lambda d)$ be used to model this interval
        • as if $\lambda$ events are expected to occur per unit time, then $\lambda d$ are expected to occur per unit $d$ of time
        • It is easy to see that $\mathbb{P}\big[\text{no events happening for time in }(t_0,t_0+d)\big]\eq\mathbb{P}\big[X'\eq 0\big]\eq e^{-\lambda d}$
      • Thus $\P{T>d}\eq e^{-\lambda d}$
        • Or: $\P{T\le d}\eq 1-\P{T>d}\eq 1-e^{-\lambda d}$
  • But this is what we'd see if $T$ followed the exponential distribution with parameter $\lambda d$
    • $\P{T\le d}\eq 1-e^{-\lambda d}$

Thus we see the time between occurrences of events in a Poisson distribution is exponentially distributed, or memoryless.

Modifications

Suppose instead $t_0$ is the start time of the process rather than the last event time, how does this change things?