(1.10) Exponential

3.1) The Exponential Distribution §

Picture this: you’re relaxing on the balcony of your penthouse on a brisk summer evening ✻. If you know that an average of 3 planes fly past your balcony per hour, and one plane just flew past, what is the expected amount of time you’ll have wait until you see the next one? In fact - how can you model the wait time between successive planes in general?

✻ The data science grind paid off.

The exponential distribution models the time between occurrences of successive events in a Poisson process. Its PMF is: $f(x)=\lambda e^{-\lambda x}$

• $\lambda$ is the rate parameter

  • It represents “how many events occur per unit time on average”
  • I.e $\text{3 planes per (one) hour}$
  • Note that it is the inverse of the mean…

• $\mu$ is the mean parameter

  • Simply the inverse; $\mu =\frac{1}{\lambda }$
  • It represents “how long between events on average”
  • I.e the 1/3 hours between each plane on average, or one plane every 20 minutes

• $E[X]=\frac{1}{\lambda }$

• Variance = $\frac{1}{{\lambda }^2}$

Q3.2) Example: Call Centre §

A customer support centre for a popular software application experiences an average of 2 calls every hour. Assuming the time between calls follows an exponential distribution, what is the probability that the next call will occur within the next 15 minutes? Also, what is the expected waiting time until the next call?

$Pr(X<\frac{1}{4})=0.393$

$E(X)=$ 30 minutes

Q3.3) What is a special property of the exponential distribution? Could you use an example to explain? §

☞ A distribution is memoryless if the probability of an event occurring in the future is not affected by how much time has already passed.

The Math $P(T>s+t\mid T>s)=P(T>t)$ $T$ = the wait time In english: The probability that it will take at least another t minutes to see the first event (RHS) is the same given that s minutes has already passed (LHS).

Real world example: If you’ve been waiting for a bus for 10 minutes, the probability that you’ll have to wait an additional 5 minutes is the same as if you just arrived at the bus stop.

Q3.4) What are other phenomena we could model using an exponential distribution? §

Category	Example
Natural Phenomena	How long will residents have to wait for the next earthquake if earthquakes occur at an average rate in a particular region?
Manufacturing	What is the duration until a machine in a manufacturing process breaks down, given a steady risk over time?
Research	How much time is expected to elapse between innovations or breakthroughs in a research field?
Electrical Engineering	How long until this electrical component fails, given that they fail at a rate of once per 5 years?
Service Operations	How much time will elapse between customer arrivals at a checkout?
Biology and Medicine	How long is the expected time between neural spikes or firings under certain conditions?
Finance	How long will it take for a financial instrument to default, given a consistent risk of default over time?
Public Transportation	What’s the expected waiting time for the next bus or train, assuming they arrive at stops at a constant average rate?
Digital Interactions	What’s the expected time between consecutive clicks or actions by a user on a website?
Insurance	How long is the expected time between successive claims from policyholders?