Simulating Random Observations

A computer can easily generate uniform random numbers on $[0,1].$ To simulate a random observation from any other continuous distribution, we use the inverse transform method:

1. Generate $u$ from $\text{Uniform}(0,1).$
2. Solve $F(x) = u$ for $x,$ where $F$ is the CDF of the target distribution.

The resulting value $x$ is a random observation from $F.$ This works because if $U \sim \text{Uniform}(0,1),$ then

$$P(F^{-1}(U) \leq x) = P(U \leq F(x)) = F(x),$$

so $F^{-1}(U)$ has CDF $F.$

Example. Let $X$ have CDF $F(x) = x^3$ for $x \in [0,1].$ To simulate $X,$ we solve $u = x^3$ for $x{:}$

$$x = u^{1/3}.$$

If the computer produces $u = 0.064,$ then the simulated observation is

$$x = (0.064)^{1/3} = 0.4.$$