I'm trying to figure out the Inverse Transform Method for stats and in my profs. slides there is this:
F(x) = cumulative dist. function, F(x) = Pr(X<=x)
For the $i$th segment $(i = 1,\dots,N)$, $$F(x) = m_i(x-x_{i-1}) + \frac{i-1}n,$$ where $$m_i = \frac{1/n}{x_i – x_{i-1}}.$$ Let $A_i$ be the inverse of $M_i$. $$F^{-1}(x) = x_{i-1} + a_i\left(x-\frac{i-1} n\right) \quad\text{for $\frac{i-1}n \le x \le \frac i n$.}$$
How do you get the inverse of $\frac{1/n}{X_i – X_{i-1}}$?
Thank you