2022/10/20 11:54 (5) IEOR E4404 001 – Ed Lessons
1: Inverse Transform
Inverse CDF for Continuous Distributions
If we recall, random variable X has continuous distribution if it has an associated density function f(x) such that for any A ⊂ R, we have:
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P(X ∈ A) = ∫ f(x)dx A
This means that the cdf F(x) = ∫ x f(y)dy, and by extension F′(x) = f(x). −∞
We can continue to use our same de nition of an inverse cdf:
F−1(u) = inf{x : F(x) ≥ u}
but since the cdf will be a continuous function, this will speci cally be equal to:
F−1(u) = inf{x : F(x) = u}
https://edstem.org/us/courses/25500/lessons/42351/preview
2022/10/20 11:54 (5) IEOR E4404 001 – Ed Lessons
Inverse Transform
Proposition Let F be the cdf of a continuous random variable, and let F −1 be the inverse as
described earlier. Let U ∼ Unif(0,1). Then P(F−1(U) ≤ x) = F(x), that is, it has cdf F(x). Proof
P(F−1(U) ≤ x) = P(F(F−1(U) ≤ F(x)) = P(U ≤ F(x))
https://edstem.org/us/courses/25500/lessons/42351/preview
2022/10/20 11:54 (5) IEOR E4404 001 – Ed Lessons
Some basic examples
We let u = xn, which gives us the inverse cdf, x = u1/n. Example 2
Let X ∼ exp(λ), then F (x) = 1 − exp(−λx) for x > 0.
f(x) = {x, 0 < x ≤ 1 (2−x), 1