Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 14 / c / i / Solution

The independence of random variables means that fixing $Y=y$ leaves the conditional probability density of $X$ equal to $f$. Given that value of $Y$, $Z$ is the translate $X+y$. HenceAt a continuity point of $f$, the expression is $f(z-y)dz+o(dz)$. Since $P(Y=y)=0$, this conditioning uses a conditional probability density, rather than division by the probability of the single event $Y=y$.