The CDF shows the probability a random variable X is found at a value equal to or less than a certain x. In other words, it’s simply the distribution function F x (x) inverted. (Of course, the simpler way is to use x = RAND("Expo")!) The UNIVARIATE procedure is used to check that the data follow an exponential distribution. The inverse distribution function (IDF) for continuous variables Fx1() is the inverse of the cumulative distribution function (CDF). The following DATA step generates random values from the exponential distribution by generating random uniform values from U(0,1) and applying the inverse CDF of the exponential distribution. This function can be explicitly inverted by solving for x in the equation F(x) = u. The exponential distribution has probability density f(x) = e –x, x ≥ 0, and therefore the cumulative distribution is the integral of the density: F(x) = 1 – e –x. To illustrate the inverse CDF sampling technique (also called the inverse transformation algorithm), consider sampling from a standard exponential distribution. This article is taken from Chapter 7 of my book Simulating Data with SAS. Output is a value or a vector of values from the exponential distribution. It can be implemented directly and is also called by the function expmemsim. Therefore, if U is a uniform random variable on (0,1), then X = F –1(U) has the distribution F. to get the inverse CDF F(-1)(u)(-log(1-u))/. The inverse CDF technique for generating a random sample uses the fact that a continuous CDF, F, is a one-to-one mapping of the domain of the CDF into the interval (0,1). This is basically the reverse of what weve just done: We start with a value between 0 and 1 (on the y-axis), and use the inverse CDF to convert that into an appropriate x-value for that particular. If you know the cumulative distribution function (CDF) of a probability distribution, then you can always generate a random sample from that distribution. Instead of evaluating the CDF at (x), we evaluate the inverse CDF at (u), where (u) is a random variable from a uniform(0, 1) distribution. However, one technique stands out because of its generality and simplicity: the inverse CDF sampling technique. There are many techniques for generating random variates from a specified probability distribution such as the normal, exponential, or gamma distribution.