The Probability Integral Transformation

Theorem 3.8.3 Probability Integral Transformation. Let X have a continuous c.d.f. F, and let Y = F (X). (This transformation from X to Y is called the probability integral transformation.) The distribution of Y is the uniform distribution on the interval [0, 1].

Corollary 3.8.1 Let Y have the uniform distribution on the interval [0, 1], and let F be a continuous c.d.f. with quantile function F −1. Then X = F −1(Y ) has c.d.f. F.

Theorem 3.8.3 and its corollary give us a method for transforming an arbitrary continuous random variable X into another random variable Z with any desired continuous distribution. To be specifific, let X have a continuous c.d.f. F, and let G be another continuous c.d.f. Then Y = F (X) has the uniform distribution on the interval [0, 1] according to Theorem 3.8.3, and Z = G− 1(Y ) has the c.d.f. G according to Corollary 3.8.1. Combining these, we see that Z = G− 1[F (X)] has c.d.f. G

computer methods for generating values from certain specified distributions see pg 171 for illustration and examples

Direct Derivation of the p.d.f. When r is One-to-One and Differentiable

Theorem 3.8.4 Let X be a random variable for which the p.d.f. is f and for which Pr(a < X < b) = 1.(Here, a and/or b can be either finite or infinite.) Let Y = r(X), and suppose that r(x) is differentiable and one-to-one for a. Let (α, β) be the image of the interval (a, b) under the function r. Let s(y) be the inverse function of r(x) for α. Then the p.d.f. g of Y is

p173