If X has the pdf X (x) = x/8, 0 < x < 4, find a

If X has the pdf ƒ_X(x) = x/8, 0 < x < 4, find a transformation Y ∼ g(X) such that Y ∼ Unif[0, 1]. The target is to achieve ƒ_Y(y) = 1 for 0 ≤ y ≤ 1. The Transformation Theorem will allow you to find h(y), from which g(x) can be obtained.

Transformation Theorem

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Let X have pdf fx(x) and let Y = g(X), where g is monotonic (either strictly increasing or strictly decreasing) on the set of all possible values of X, so it has an inverse function X = g-¹(Y) = h(Y). Assume that h has a derivative h' (y). Then fy(y) = fx (h(y)) |h' (y)] (4.12)