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

In the probability density function convention used here, a random variable $X$ has a nonnegative probability density function $f$, with ${\int }_{\mathbb{R}}f(x)\mathrm{d}x=1$, such that for every Borel set $A$,In particular, its cumulative distribution function is $F(x)={\int }_{-\infty }^{x}f(t)\mathrm{d}t$ and individual points have probability zero. More precisely, this property is the statement that the distribution is an absolutely continuous probability distribution; continuity of the cumulative distribution function alone is a weaker property and need not imply a probability density function.