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 / a

The cumulative distribution function isFor an absolutely continuous probability distribution, the probability density function is a nonnegative integrable function $f_X$ withThus interval probabilities are integrals of $f_X$, and $F_X^{\prime }=f_X$ almost everywhere, in particular at continuity points of $f_X$.

The word “continuous” in the source is understood in this density sense. A merely continuous cumulative distribution function need not possess a probability density function; it can instead describe a singular continuous distribution.

For an absolutely continuous probability distribution, its expectation isThis is finite when ${\int }_{-\infty }^{\infty }|x|f_X(x)dx<\infty$. More generally it is an extended real value if only one of the positive and negative parts diverges; it is undefined if both diverge. For nonnegative $X$ it always exists in $[0,\infty ]$.

Let $S(x)=\mathbb{P}(X>x)=1-F_X(x)$ for $x\ge 0$. Since $f_X=0$ on the negative half-line, $X\ge 0$ almost surely. We have $S^{\prime }=-f_X$ almost everywhere. On a finite interval, integration by parts givesThe lower endpoint contributes zero. The stated hypothesis gives $RS(R)\to 0$ as $R\to \infty$. Taking the increasing limits of the two nonnegative integrals proves the tail-integral identityThe equality allows the value $+\infty$; the boundary hypothesis alone does not force a finite expectation. In fact the formula for nonnegative $X$ also follows directly from Tonelli theorem, without that additional boundary hypothesis, by integrating $x={\int }_0^{\infty }{1}_{\{t<x\}}dt$ against the probability density function.