Non-uniform random distributions

In th eprevious section we learned how to generate random deviates with a uniform probability distribution in an interval $[a,b]$. This distributioon is normalized, so that

$$\int _a^b {P(x)dx}=1.$$

Hence, $P(x)=1/(b-a)$.

Now, suppose that we generate a sequence $\{x_i\}$ and we take some function of it to generate $\{y(x_i)\}=\{y_i\}$. This new sequence is going to be distributed according to some probaility density $P(y)$, such that

$$P(y)dy=P(x)dx$$

or

$$P(y)=P(x)\frac{dx}{dy}.$$

If we want to generate a desired normalized distribution $P(y)$, we need to solve the differential equation:

$$\frac{dy}{dy}=P(y).$$

But the solution of this is

$$x=\int _0^y {P(y')dy'}=F(y).$$

Therefore,

$$y(x)=F^{-1}(x),$$ (95)

where $F^{-1}$ is the inverse of $F$.