Poisson Distribution & Exponential Distribution

Investigating Poisson and Exponential Distributions: Time, Events, and Probability Density Functions

Poisson distribution is for discrete variable while Exponential distribution is for continuous. However, both distributions are using one single parameter λ to specify the distribution.

In fact, the λ using in both distributions are actually referring to the same thing which is the average number of events in a unit time.

For Poisson, X refers to the number of events in the time interval. For Exponential, X refers to the waiting time until the first occurrence of the event.

For Poisson process, we got

Since X indicates the waiting time until the first occurence of the event,

We then can derive the PDF from the CDF.