|Probability density function
|Cumulative distribution function
|Parameters||λ > 0 rate, or inverse scale|
|Support||x ∈ [0, ∞)|
|CDF||1 − e−λx|
|Median||λ−1 ln 2|
|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 u=−x2 and v=e−λx. From this we have du=−2x and dv=−λe−λx
We see that the integral is just E[X] which we solved for above.