Transformation of random variable ended up in a negative pdf

Question

My question is about the result a a transformation of a random variable (rv). We may end up with a negative pdf after a transformation. Suppose that $f_x(X) = \frac{2}{x^3}$, x > 1 and Y = g(X) = 1/X. So X = h(y) = 1/Y. Then

$F_Y(Y) = F_X(h(y))= \int_1^{h(y)}2x^{-3} = \int_1^{1/y}2x^{-3} = -x^{-2}|^{1/y}_1 = 1 - y^2$

In this case:

$\frac{dF_Y}{dY} = -2y$

So the pdf of Y=1/X is negative. What is wrong here?

Answer 1

Your formula is true only of $h$ is increasing.

Note that to $Y<y$ transforms to $x>h(y)$. So $$ F_Y(y) =1- F_X(h(y))= 1-\int_1^{h(y)}2x^{-3} = 1- \int_1^{1/y}2x^{-3} = 1-\left(-x^{-2}|^{1/y}_1 \right)= y^2$$ The above result is true for $y\le 1$ since $$0 \le Y \le 1$$