Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 9 / a / i / Solution

The cumulative distribution function is $F_X(x)=\mathbb{P}(X\le x)$. In the absolutely continuous setting of a probability density function, the probability density function is a nonnegative integrable function $f_X$ satisfyingThus $\mathbb{P}(a<X\le b)={\int }_a^b{f}_X(t)dt$ and $F_X^{\prime }=f_X$ almost everywhere. A merely continuous cumulative distribution function need not possess a probability density function; the density assumption means absolute continuity.