Consider the Random Variable $X $ given by PDF $f (x)=0$ if $x\le 0$, $f (x)=1/2$ if $0\le x\le 1$, and $f (x)=1/(2x^2)$ if $1<x\le\infty$. I wish to determine the cumulative distribution function $F_Y$ of the RV $Y=1/X$.
I can find out Pr $(Y\le x)$ if $x>0$ but am confused how to proceed when $x <0$. It seems to me Pr $(Y\le x)$ for $x<0$ is $1-F_X(1/x)=1$ but this is wrong as $F_X$ is no longer increasing.
$f(x)=0$ for $x \leq 0$ means that $X$ is a positive random variable. Hence $Y$ is also positive, so $P\{Y\leq x\}=0$ if $x <0$.