Exponential Distribution

Probability density function
Cumulative distribution function
Parameters λ > 0 rate, or inverse scale
Support x ∈ [0, ∞)
PDF λ e−λx
CDF 1 − e−λx
Mean λ−1
Median λ−1 ln 2
Mode 0
Variance λ−2
Skewness 2
Ex. kurtosis 6
Entropy 1 − ln(λ)

Exponential distribution refers to a statistical distribution used to model the time between independent events that happen at a constant average rate λ. Some examples of this distribution are:

  • The distance between one car passing by after the previous one.
  • The rate at which radioactive particles decay.

For the stochastic variable X, probability distribution function of it is:


and the cumulative distribution function is:


Exponential distribution is denoted as  , where m is the average number of events within a given time period. So if m=3 per minute, i.e. there are three events per minute, then λ=1/3, i.e. one event is expected on average to take place every 20 seconds.



We derive the mean as follows.


We will use integration by parts with u=−x and v=e−λx. We see that du=-1 and dv=−λe−λx.




We use the following formula for the variance.


We'll use integration by parts with   and  . From this we have   and  .


We see that the integral is just   which we solved for above.