Inverse CDF Method

In our CDF we have $F(x)=P(X\le x)=p$ with $0\le F(x)\le 1$.

We take the inverse $F^{-1}(p)$. This is value $x$ such that $P(X\le x)=p$.

We can think of about this in terms of quantiles.

For example,

• $0.5$ is the median
• $0.25$ is the first quartile And so on.

Think of the Inverse CDF as the quantile function.

Inverse Transform Sampling

Suppose we draw some Random Variable $U$ from a uniform distribution on the interval $(0,1)$.

We set take $F^{-1}(U)$. Now, $F$ will return a value $X$ associated with the quantile value. We can say that $X$ follows the distribution of $F$.