### 3.2.1 Sampling via Inversion of the cdf

Since the r.v. x and the cdf are 1-to-1, one can sample x by first sampling and then solving for x by inverting , or . But Eq. (83) tells us that the cdf is uniformly distributed on [0,1], which is denoted . Therefore, we simply use a random number generator (RNG) that generates numbers, to generate a sample from the cdf . Then the value of x is determined by inversion, . This is depicted graphically in Figure 12. The inversion is not always possible, but in many important cases the inverse is readily obtained.

Figure 12: Sampling Using the Inverse of the cdf.

This simple yet elegant sampling rule was first suggested by von Neumann in a letter to Ulam in 1947 [Los Alamos Science, p. 135, June 1987]. It is sometimes called the ``Golden Rule for Sampling''. Since so much use will be made of this result throughout this chapter, we summarize below the steps for sampling by inversion of the cdf:

Step 1.
Sample a random number from
Step 2.
Equate with the cdf:
Step 3.
Invert the cdf and solve for x: