Statistics/Distributions/Exponential
Exponential Distribution[edit]
Probability density function 

Cumulative distribution function 

Parameters  λ > 0 rate, or inverse scale 

Support  x ∈ [0, ∞) 
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) 
MGF  
CF 
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.
Mean[edit]
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}.
Variance[edit]
We use the following formula for the variance.
We'll use integration by parts with u=−x^{2} 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.