Exponential Distribution

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

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

However, 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}. \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). We define the density to zero for x<0.

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

I plot the Probability Density Functions and the corresponding Cumulative Density Functions to the right. Here, I use 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

The SAS code below lets you set \sigma and draw the probability density function for the corresponding exponential function. I encourage you to play around with this code. 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);
/* 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);

Finally, check out the related pages about the Lognormal and Gamma Distribution in SAS.