Exponential

The exponential cumulative distribution function with parameter μ, or mean μ,is defined by the integral

$\displaystyle \func$ExponentialDist$\displaystyle \left(\vphantom{ x;\mu }\right.$x;μ$\displaystyle \left.\vphantom{ x;\mu }\right)$ = $\displaystyle {\frac{{1}}{{\mu }}}$$\displaystyle \int_{{0}}^{{x}}$e$\scriptstyle {\frac{{-u}}{{\mu }}}$du =  1 - e$\scriptstyle {\frac{{-x}}{{\mu }}}$

of the density function

$\displaystyle \limfunc$ExponentialDen(u;μ) = $\displaystyle {\frac{{1}}{{\mu }}}$e$\scriptstyle {\frac{{-u}}{{\mu }}}$

for x≥0, and is 0 otherwise.

The inverse exponential distribution function

$\displaystyle \limfunc$ExponentialInv$\displaystyle \left(\vphantom{ \alpha ;\mu }\right.$α;μ$\displaystyle \left.\vphantom{ \alpha ;\mu }\right)$ = μln$\displaystyle {\frac{{1}}{{%
1-\alpha }}}$

is the value of x for which the  integral has the value   α, as illustrated by the following:
$\displaystyle \limfunc$ExponentialInv$\displaystyle \left(\vphantom{ .73;.58}\right.$.73;.58$\displaystyle \left.\vphantom{ .73;.58}\right)$ = .58 ln$\displaystyle {\frac{{1}}{{1-.73}%
}}$ = .75941  
$\displaystyle \func$ExponentialDist$\displaystyle \left(\vphantom{ .75941;.58}\right.$.75941;.58$\displaystyle \left.\vphantom{ .75941;.58}\right)$ = 1 - e$\scriptstyle {\frac{{-.75941}}{{.58}<tex2html_comment_mark>53}}$ = .73  

The following plots show distribution functions $\func$ExponentialDist$\left(\vphantom{ x;\mu }\right.$x;μ$\left.\vphantom{ x;\mu }\right)$ and density functions $\func$ExponentialDen(x;μ), for the parameters μ = 1, 3, 5  and -2≤x≤4.

itbpFU2.1577in1.7573in0inExponential density functions itbpFU2.1577in1.7573in0inExponential distribution functions