Statistics/Distributions/Exponential

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:

$f_x(x)=\{\begin{array}{cc}\lambda e^{-\lambda x},& \text{if }x\ge 0\\ 0,& \text{if }x<0\end{array}$

and the cumulative distribution function is:

$F_x(x)=\{\begin{array}{cc}0,& \text{if }x<0\\ 1-e^{-\lambda x},& \text{if }x\ge 0\end{array}$

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