Uniform and Chi-square distributions I am trying to solve this problem:
Let $X \sim U[0,1]$. Then $Y = -2\log X$ is $\chi$-square distribution with parameter $2$.
I proceeded this way,
$$P(Y \leq y) = P(-2\log X \leq y) = P(X \leq \exp(-y/2))$$
$$F_X(x) = \frac{x-a}{b-a},\quad a \leq x <b$$
$$\Rightarrow F_X(\exp(-y/2)) = \exp(-y/2),\quad 0 \leq x <1.$$
Now I found the PDF:
$$f(y) = -1/2\exp(-y/2)$$
But the $\chi$-square PDF with parameter $2$ is given by
$$f(y) = 1/2\exp(-y/2)$$
I am not sure why I get the negative sign. Any thoughts?
Thanks

the negative sign will be canceled because we have to the absolute value of the derivative according to The Jacobian in one-to-one transformation technique.
