# Statistics/Distributions/Exponential

## Contents

### Exponential Distribution

Parameters Probability density function Cumulative distribution function λ > 0 rate, or inverse scale x ∈ [0, ∞) λ e−λx 1 − e−λx λ−1 λ−1 ln 2 0 λ−2 2 6 1 − ln(λ) $\left(1 - \frac{t}{\lambda}\right)^{-1}\,$ $\left(1 - \frac{it}{\lambda}\right)^{-1}\,$

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:

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

and the cumulative distribution function is:

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

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

#### Mean

We derive the mean as follows.

$\operatorname{E}[X] = \int^\infin_{-\infin} x \cdot f(x) dx$
$\operatorname{E}[X] = \int^\infin_{0}x\lambda e^{- \lambda x} dx$
$\operatorname{E}[X] = \int^\infin_{0}(-x)(-\lambda e^{- \lambda x}) dx$

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

$\operatorname{E}[X] = \left[-x \cdot e^{- \lambda x}\right]^\infin_{0} - \int^\infin_{0}(e^{- \lambda x})(-1) dx$
$\operatorname{E}[X] = [0-0] + \left[{-1 \over \lambda}(e^{ -\lambda x})\right]^\infin_{0}$
$\operatorname{E}[X] = \left[0-{-1 \over \lambda}\right]$
$\operatorname{E}[X] = {1 \over \lambda}$

#### Variance

We use the following formula for the variance.

$\operatorname{Var}(X) = \operatorname{E}[X^2]-(\operatorname{E}[X])^2$
$\operatorname{Var}(X) = \int^\infin_{-\infin} x^2 \cdot f(x) dx-\left({1 \over \lambda} \right)^2$
$\operatorname{Var}(X) = \int^\infin_{0}x^2 \lambda e^{- \lambda x} dx-{1 \over \lambda^2}$

We'll use integration by parts with u=−x2 and v=e−λx. From this we have du=−2x and dv=−λe−λx

$\operatorname{Var}(X) = \left\{\left[-x^2 \cdot e^{- \lambda x}\right]^\infin_{0} - \int^\infin_{0}(e^{- \lambda x})(-2x) dx\right\}-{1 \over \lambda^2}$
$\operatorname{Var}(X) = [0-0]+ {2 \over \lambda}\int^\infin_{0}x \lambda e^{- \lambda x} dx -{1 \over \lambda^2}$

We see that the integral is just E[X] which we solved for above.

$\operatorname{Var}(X) = {2 \over \lambda}{1 \over \lambda} -{1 \over \lambda^2}$
$\operatorname{Var}(X) = {1 \over \lambda^2}$