The Exponential distribution models the waiting time until the next “arrival” (occurrence) of some event.
- The next text message arriving on your phone
- The next car accident arriving in Philly
This distribution is closely related to the Poisson distribution, which models the number of arrivals (successes) in a fixed time period. It is also Memoryless.
Example: motivation
Assume there are 17 major earthquakes per year. Let be the number of major earthquakes in the next year. Then, and so .
Let be the number of major earthquakes in the next years. Then, , where . Let be the waiting time from now until the next major earthquake. Then, for all in , the CDF of is
We can further expand , and taking the derivative we can find the PDF to be , for support .
PDF and CDF
A Random variable has the Exponential distribution with rate parameter if the PDF of is
for support , and zero otherwise. The textbook writes . Note that the here is different from in the Poisson.
The CDF is given by
Standard Exponential distribution
Just like the standard Normal and standard Uniform, we can let and have .
We have and . For a proof of the variance, see the example for the first theorem under Usefulness.
Using a scale transformation
To go between and , we can simply divide by and multiply by , respectively. If , then we have .
To prove, let . Then, we have CDF . This will be true as long as , the support.
- Then, to go to , multiply the r.v. by .
- To go to , divide the r.v. by .
MGF
Standard Exponential
The MGF of the standard exponential distribution can be found using LOTUS, where and . We have
To guarantee that the integral remains finite, we set . So the open interval is given by .
General
For a r.v. , we have .
General expectation and variance
Using the scale transformation property from above, we can derive the Expectation and Variance for a r.v. . We have and .