Transformation of non-monotonic density functions

[island-boy]

What's the general way to solve them?
I know that the thm which states:
f(y) = f(g(y)) * |D g(y)| is not applicable, since it is only applicable to strictly increasing or decreasing functions.

here's a sample question: Let X have the density function f(x) = 1/2 for -1<x<1, 0 otherwise. Let Y = X + 1 for 0<x<1 and Y = -2X for -1<x<0. Find the density function of Y.

If I use the above forumule, the domains for Y = X+1 and Y =-2X would intersect while having different density functions...so I'm not sure what to do there.

Thanks

 

[lippyka]

for any help!The general way to solve these types of problems is to use the change of variables formula. This states that, given a function y = f(x) with an associated density function f(x), the density function of the new variable y can be found using the formula: f_y(y) = f_x(f^{-1}(y)) * |J| where J is the Jacobian of the transformation and f^{-1} is the inverse of the function f(x). In your given example, the transformation is y = X + 1 for 0<x<1, and y = -2X for -1<x<0. To apply the formula, you need to find the inverse of this transformation (i.e. x = f^{-1}(y)), then find the Jacobian of the transformation. Once you have this, you can plug in the values into the formula to get the desired density function of Y.