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.


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 .


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 .


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 .