I have a hard time to discrete the gamma distribution . Can anyone help me? Thanks in advance.

The travel time from A to B denoted as X follows a contituous gamma distribution. I want to discretize the CDF of X into λ intervals with an equal cumulative probability. For example to divided the CDF into 20 intervals (0, 0.05, 0.1, ..., 0.95, 1). A mean value xi is needed to calculated in each interval, P(X= xi) = ω ,i= 1,2,...,λ. My problem is how to calculate xi for each interval?

Denote a set of intervals to be (iω-ω, iω), i=1,2,…, λ, where ω is the cumulative probability of an interval, 0<ω<1 and λω=1.

According to the Mean-Value Theorem, within each interval (iω-ω, iω) there is at least one point xq satisfies:

f(xi) =ω/{v(iω) − v(iω−ω)}

where f(xi) is the PDF of X at point xq, and v(iω −ω ) and v(iω) are the inverse of CDF at the boundary points of the interval.

(1) how to inverse the CDF at iω ?

(2) how to inverse the PDF at f(xi) ?