Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 26K (Probability and Measure)

Let ${\mu }_X$ and ${\mu }_Y$ be the probability laws of $X$ and $Y$. The hypothesis saysfor every compactly supported continuous function $f$. The theorem that compactly supported continuous functions determine a finite Borel measure gives ${\mu }_X={\mu }_Y$. Hence $X$ and $Y$ have the same law.

The integrands are absolutely integrable because $f$ has compact support, $g\in L^1(\mathbb{R})$, and $|{\phi }_Z|\le 1$. Fubini's theorem therefore permits us to insert ${\phi }_Z(s)=\mathbb{E}[e^{isZ}]$ and interchange the expectation and integrals:With $r=\epsilon s$, the inner integral in $s$ isNow set $t=(Z-x)/\epsilon$. The change of variables $x=Z-\epsilon t$ giveswhich is the required identity.

Choose any $g$ with $g,\hat{g}\in L^1$, $g(0)\ne 0$, and for which Fourier inversion theorem applies, for example a Gaussian function. The left side factors through the Fourier transform of $f$ asOn the right side, $f(Z-\epsilon t)\to f(Z)$ andThe dominated convergence theorem and Fourier inversion at zero giveConsequently the Fourier-cutoff recovery of expectations from a characteristic function isThe right side depends only on ${\phi }_Z$. Equal characteristic functions therefore give equal expectations for every compactly supported continuous $f$, and part a yields the uniqueness theorem for characteristic functions.

Choose $M<\infty$ such that $|X|,|Y|\le M$ almost surely. Equality of the positive moments, together with equality of the zeroth moments $\mathbb{E}[1]=1$, givesfor every polynomial $p$. By the Weierstrass approximation theorem, every continuous function on $[-M,M]$ is a uniform limit of polynomials. Expectations preserve this limit, so $X$ and $Y$ have equal expectations for every continuous function on their common compact support. The moment determinacy on a compact interval therefore gives equality of their laws.

SetThese are bounded random variables with values in $(0,1]$, and for every positive integer $n$,Part c shows that $U$ and $V$ have the same law. Since $x\mapsto e^{-x}$ is injective on $[0,\infty )$ with inverse $u\mapsto -\log u$, taking the pushforward measure by this inverse shows that $X$ and $Y$ have the same law. This proves the uniqueness theorem for Laplace transforms of nonnegative random variables.

The displayed density is that of a gamma distribution with shape $k+1$ and rate one. PutBy the additivity of independent gamma distributions with a common rate,Equivalently, the independence of random variables makes the Laplace transform of a nonnegative random variable of the sum equalTherefore the required probability density function is