／var／logmarcus chiu

Poisson vs Exponential

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comparing:

Deriving Exponential Distribution From Poisson Distribution

Consider the sequence of rare events, where the number of occurrences during time 𝑡 has Poisson distribution with a parameter 𝜆 inversely proportional to 𝑡 (i.e. 𝑡 = 1/𝜆)

Given a mean number of events (𝜆) that happen within unit time, the number of events (𝑥) occurring within that unit time has Poisson Distribution and can be described using Poisson’s probability mass function

  • 𝐏(𝑋=𝑥) = 𝑒−𝜆(𝜆𝑥/𝑥!) for 𝑥 = 0, 1, 2, …
PROBLEM

We want to show that the times between rare events are Exponential, using only Poisson Distribution

SOLUTION

Consider event 𝐴:

  • 𝐴 = “the time 𝑇 until the next event occurring is greater than 𝑡”
  • 𝐴 = “no events occur within the time interval [0, 𝑡]”

now consider random variable 𝑋 which is the number of events during the time interval [0, 𝑡].

This 𝑋 has Poisson distribution with parameter 𝜆’=𝜆𝑡:

  • 𝜆 = average number of events happening per time unit
  • 𝑡 = number of time units
  • 𝜆’ = 𝜆𝑡 = average number of events happening in time interval [0, 𝑡]

Thus, the probability of 0 events occurring during the time interval [0, 𝑡] is described below

  • 𝐏(𝑋=0) = 𝑒-𝜆𝑡 [(𝜆𝑡)0 / 0!]
  • 𝐏(𝑋=0) = 𝑒-𝜆𝑡

This probability is equivalent to the probability of event 𝐴 happening

  • 𝐏(𝑋 = 0) = 𝐏{event 𝐴 occurs}

Then we can compute the 𝐶𝐷𝐹 of 𝑇 as

  • 𝐶𝐷𝐹𝑇(𝑡) = 1 - 𝐏(𝑇 > 𝑡)
  • 𝐶𝐷𝐹𝑇(𝑡) = 1 - 𝐏(event 𝐴 occurs)
  • 𝐶𝐷𝐹𝑇(𝑡) = 1 - 𝐏(𝑋 = 0)
  • 𝐶𝐷𝐹𝑇(𝑡) = 1 - 𝑒-𝜆𝑡

And here we recognize the Exponential 𝐶𝐷𝐹. Therefore, the time until the next arrival has Exponential distribution

Take the derivative of 𝐶𝐷𝐹𝑇(𝑡) to obtain the exponential probability density function

  • 𝑝𝑑𝑓𝑇(𝑡) = 𝐶𝐷𝐹𝑇’(𝑡)
  • 𝑝𝑑𝑓𝑇(𝑡) = (1 - 𝑒-𝜆𝑡)’
  • 𝑝𝑑𝑓𝑇(𝑡) = 0 - (-𝜆𝑒-𝜆𝑡)
  • 𝑝𝑑𝑓𝑇(𝑡) = 𝜆𝑒-𝜆𝑡