I’d like to create a place to store all my math fails, so i can remember them and learn to never repeat them again.

2026-02-01

\[f_X (x) = 1_{(0,1)}(x), Y = (1 - X)^{-1/3} \rightarrow f_Y (y)= ?\]

I set the transformation

\[h(x) = (1-x)^{-1/3} \rightarrow h^{-1}(x) = 1 - x^{-3}\] \[f_Y = (f_X \circ h^{-1}(y)) \cdot \frac{dh^{-1}(y)}{dy}\]

Since \(f_X = 1_{(0,1)}\) then \(f_Y = f_X \circ h^{-1}\). I assumed algebraly that \(f_Y(y) = 1_{(0,1)}\bigl(h^{-1}(y)\bigr)\)

I wrongly assumed that the bounds of 1 don’t depend on y and just did $f_Y = f_X$.