Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 9 / b / ii / Solution

Reversing sign reverses the inequality. Since $X$ has a probability density function, it has no point masses, soDifferentiating the cumulative distribution function givesalmost everywhere: the minus sign outside $F_X$ cancels the minus sign from the chain rule. If point masses were allowed, the cumulative distribution function would instead require the left limit $1-F_X((-y{)}^{-})$.