As the title says, we are given a random variable $X$ whose distribution function $F_{X}$ is known. How can we determine the distribution of $Y = \max\{X,0\}$? Here is what I have tried: \begin{align*} \mathbb{P}(Y\leq y) & = \mathbb{P}(\max\{X,0\} \leq y)\\\\ & = \mathbb{P}((\max\{X,0\} \leq y)\wedge(X\geq 0)) + \mathbb{P}((\max\{X,0\}\leq y)\wedge(X < 0))\\\\ & = \mathbb{P}(0\leq X \leq y) + \mathbb{P}(X < 0\leq y)\\\\ & = \mathbb{P}(0\leq X \leq y) + \mathbb{P}(X < 0)\\\\ & = F_{X}(y) - F_{X}(0^{-}) + F_{X}(0^{-})\\\\ & = F_{X}(y). \end{align*}
and then I get stuck as I do not know what to do with the second term from the above expression.
I can see that, if $y < 0$, then $\mathbb{P}(Y\leq y) = 0$. But what if $y\geq 0$?