Exponential Distribution

The Exponential distribution is a continuous probability distribution. It models the time between events, which on average occur at a constant, i.e. a Poisson process, where events occur continuously and indepently. It has Probability Density Function

    \begin{equation*} f(x) = \frac{1}{\sigma} \exp \left(- \frac{x}{\sigma} \right), \quad x \geq 0. \end{equation}

Though often you will see the density defined as

    \begin{equation*} f(x) = \lambda e^{- \lambda x}, \quad x \geq 0. \end{equation}

where \lambda=\frac{1}{\sigma}. In the above \sigma is the scale parameter and \lambda is the rate parameter. The rate parameter indicates the rate at which the event occurs. The two densities are the same, but since the SAS PDF function takes \sigma as argument, I like to go with that one. If a stochastic variable X is exponentially distributed, we write X \sim Exp(\sigma). The density is defined to be zero for x<0.

The Exponential is a special case of the Gamma distribution with shape parameter a=1 and scale parameter \lambda. Also the Exponential distribution can be interpretted as the

To the right, I have plotted Probability Density Functions and the corresponding Cumulative Density Functions for Exponential Distributions with different values of \sigma. You can download the code creating these plots here.

Exponential Probability Density Function PDF
Exponential Cummulative Density Function CDF
SAS Code Example

Below, I have written a small SAS program, that lets you set \sigma and draw the probability density function for the corresponding exponential function. I encourage you to play around with this code to familiarize yourself with the impact of \sigma on the shape of the density.

/* Exponential PDF Curves */
%let sigma = 1;
 
data Exponential_PDF;
         do x=0 to 4 by 0.01;
            Exponential_PDF = pdf('Exponential', x, &sigma);
            output;
         end;
run;
 
/* Draw PDF Curve */
title "Exponential Density For (*ESC*){unicode sigma} = &sigma";
proc sgplot data=Exponential_PDF noautolegend;
   series x=x y=Exponential_PDF / lineattrs = (thickness=2 color=black) legendlabel="Exponential PDF";
   keylegend / position=NE location=inside across=1 noborder valueattrs=(Size=12 Weight=Bold) 
                                 titleattrs = (Size=12 Weight=Bold);
   xaxis label='x' labelattrs = (size=12 weight=Bold);
   yaxis display=(nolabel) label='PDF' labelattrs = (size=12 weight=Bold);
run;
title;