# Solved – Reason for absolute value of Jacobian determinant in change-of-variable formula

When we have a random variable $$x$$ with a probability density $$p(x)$$, and a function $$y = f(x)$$ that is differentiable and can be solved for $$x = g(y)$$, the change of variable formula leads us to a density for $$y$$ given by

$$p(x) , dx = p(x) left| g'(y) right| , dy = p(x) left| frac{1}{f'(x)} right| , dy = p(x) left| frac{dx}{dy} right| , dy$$
where $$frac{dx}{dy}$$ is called (to my knowledge even in the univariate case) the Jacobian of the transformation (as in Zill & Wright, p. 792). In general this would be a determinant of a Jacobian matrix $$mathbf{J}(mathbf{g}(mathbf{y}))$$, obviously. But I never understood why does it enter in absolute value? I have read somewhere that it's because $$f(x)$$ could have a negative derivative whereas probabilities are confined to be positive, but that sounds more like a post-hoc justification than a mathematical result. Is there a way to derive this fact?

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For a specific example, in addition to @whuber's advice, let $$y=f(x)=-2x$$, and $$x=g(y)=-y/2$$; and $$x in [0,1]$$, i.e. the support. Then, $$y$$ would be in the range $$[-2,0]$$. Also, we have $$g'(y)=-1/2, f'(x)=-2$$.
Normally, you'd take the integral $$int p(g(y))left|frac{dx}{dy}right|dy$$ from $$-2$$ to $$0$$, while using the formula. However, it actually is from $$0$$ to $$-2$$, since $$x$$ and $$y$$ directions differ, i.e. $$int_{0}^{-2} p(g(y))frac{dx}{dy}dy=int_{-2}^{0} p(g(y))left(-frac{dx}{dy}right)dy=int_{-2}^{0} p(g(y))left|frac{dx}{dy}right|dy$$
The use of absolute value removes the need of considering the inverse directions (i.e. negative directions of $$x$$ and $$y$$ which is reflected by negative derivatives).