Poisson vs Exponential distributions

Related yet different, here’s how…

A quick note on the “preliminary terrors” of notation:

  • e is Euler’s number – you’ll find the e on your calculator or the EXP() function in Excel
  • The parameter is conventionally written as λ (pronounced lambda).



Number of events that occur in an interval of time Time taken between 2 events occurring
For example… the number of Metrorail trains that arrive at the platform in an hour For example… the time between one Metrorail train arriving and the next
6 trains 10 minutes between trains
The random variable is discrete – this is a countable number of things that happen The random variable is continuous – typically time, but may represent other factors like distance
l represents the number of events (per unit of time) l represents time between individual events
Graphically, a Poisson distribution looks like this:


Graphically, an Exponential distribution looks like this:


Calculate probability as follows:

P(X=x) = eλλx / x!

Calculate probability as follows:

P(x>n) = eλ(n)

P(x<n) = 1 – eλ(n)

P(n<x<m) = P(x<m) – P(x<n)

On average 6 trains arrive at the platform per hour…

What is the probability that fewer than 2 trains arrive in the next 20 minutes? What is the probability that a train arrives every 8 minutes or less?
λ = 2 per 20 minutes λ = 0.1 per minute (6/60)
P(X<2) = P(X=0) + P(X=1)

= e220 / 0!  +  e21/ 1!

= 0.41

P(x<8) = 1 – e0.1(8)

= 0.45


2 amazingly simple video tutorials are available on YouTube if you need a quick walk-through:

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