Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 6

The cumulative distribution function isThe total integral of the nonnegative probability density function is one.

For a random variable with a probability density function, a singleton has zero probability:Equivalently, its cumulative distribution function is continuous, so it has no jump at $x$.

The integers form a countable set, and the events $\{X=k\}$ are disjoint. By countable additivity and part ii,Countability is essential: one cannot use the same sum argument over an arbitrary uncountable collection of zero-probability events.

Normalization of the probability density function requiresHence $\boxed{c=t-1}$. The distribution is a Pareto distribution with minimum one and shape parameter $t-1$.

The condition $t>2$ guarantees a finite first moment. Direct integration gives

For an integer $k\ge 1$, $\lfloor X\rfloor \ge k$ is the same event as $X\ge k$. Integration for the Pareto distribution givesThe second identity subtracts two successive tail probabilities, equivalently integrating over $[k,k+1)$. If $\mathbb{N}$ includes zero, the extra case is $\mathbb{P}(\lfloor X\rfloor \ge 0)=1$ and $\mathbb{P}(\lfloor X\rfloor =0)=0$, since $X\ge 1$ almost surely.

For a nonnegative integer-valued random variable $W$, the tail-sum formula states $\mathbb{E}[W]={\sum }_{k=1}^{\infty }\mathbb{P}(W\ge k)$. It follows by writing $W={\sum }_{k\ge 1}{1}_{\{W\ge k\}}$ and using nonnegativity to interchange sum and expectation.

With $W=\lfloor X\rfloor$, this givesThe Riemann zeta function is finite here because $t-1>1$.

By part iii, $X$ is an integer with probability zero. Thus $\lceil X\rceil =\lfloor X\rfloor +1$ almost surely, andThe pointwise inequalities $\lfloor X\rfloor \le X\le \lceil X\rceil$ implyPutting $s=t-1>1$ and rearranging yieldsThis extracts bounds for an infinite series from simple rounding inequalities.