1. The Exponential distribution
Just as there are many named discrete distributions, there are many named continuous distributions as well. For example, let's take another trip to Poissonville and stand at a bus stop. We know that the number of buses that will arrive per hour are Poisson distributed. But the amount of time between arrivals of buses is Exponentially distributed.
2. The Exponential distribution
The Exponential distribution has this story: the waiting time between arrivals of a Poisson process are exponentially distributed. It has a single parameter, the mean waiting time. This distribution is not peaked,
3. The Exponential PDF
as we can see from its PDF. As an example,
4. Possible Poisson process
As an example, we can look at the time between all incidents involving nuclear power since 1974. We might expect incidents to be well-modeled by a Poisson process, i.e., the timing of one incident is independent of all others. So, the time between incidents should be Exponentially distributed.
5. Exponential inter-incident times
We can compute and plot the CDF we would expect based on the mean time between incidents and overlay that with the ECDF from the real data. We take our usual approach where we draw many samples out of the Exponential distribution, using the mean inter-incident time as the parameter. We make the plot and label the axes.
6. Exponential inter-incident times
We see that it is close to being Exponentially distributed, indicating the nuclear incidents can indeed be modeled as a Poisson process.
The Exponential and Normal are just two of many examples of continuous distributions. Importantly, in many cases you can just simulate your story to get the CDF. Remember, you have the power of a computer. If you can simulate a story, you can get its distribution!
7. Let's practice!
Now, let's play with the Exponential distribution in some exercises.