Introduction to the Mathematical and Statistical Foundations of Econometrics

Transformations of Absolutely Continuous Random Variables

If X is absolutely continuously distributed, with distribution function F(x) = /Е f (u)du, the derivation of the distribution function of Y = g(X) is less trivial. Let us assume first that g is continuous andmonotonic increasing: g(x) < g(z) if x < z. Note that these conditions imply that g is differentiable.1 Then g is a one-to-one mapping - that is, for each y є [g(—to), g(oo)] there exists one and only one x є К U {-to} U {to} such that y = g(x). This unique x is denotedby x = g-1( y). Note that the inverse function g-1( y) is also monotonic increasing and differentiable. Now let H(y) be the distribution function of Y. Then

H(y) = P(Y < y) = P(g(X) < y)

= P(X < g-1(y)) = F(g-1(y)). (4.15)

Except perhaps on a set with Lebesgue measure zero.

Taking the derivative of (4.15) yields the density H(y) of Y:

h( y) = H (y) = f(g-1(y))^^- ■ (4.16)

dy

If g is continuous and monotonic decreasing: g(x) < g(z) if x > z, then g-1(y) is also monotonic decreasing, and thus (4.15) becomes

H(y) = P(Y < y) — P(g(X) < y)

— P(X > g-1(y)) = 1 - F(g-1(y)),

Note that in this case the derivative of g-1(y) is negative because g-1(y) is monotonic decreasing. Therefore, we can combine (4.16) and (4.17) into one expression:

Theorem 4.2: If X is absolutely continuously distributed with density f, and Y — g(X), where g is a continuous, monotonic real function on R, then Y is absolutely continuously distributed with density h(y) given by (4.18) if min[g(—TO), g(TO)] < y < max[g(—TO), g(TO)], and h(y) — 0 elsewhere.