Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 28L (Principles of Statistics)

Let $Q=F^{-1}$ be the quantile function. For $p\in (0,1)$,The reverse implication is immediate from the infimum defining $Q$. For the forward implication, if $Q(p)<t$ there is a point $s\le t$ with $F(s)\ge p$. If $Q(p)=t$, choose $s_m<t+1/m$ with $F(s_m)\ge p$; monotonicity and right continuity giveThe endpoint values $p=0,1$ have zero probability under a continuous uniform variable.

ConsequentlyThus the inverse transform sampling identity is

Choose an initial value $X_0$. One sweep of the Gibbs sampler consists ofAfter a burn-in period, the pairs $(Y_m,X_m)$ are retained as approximate samples. Standard irreducibility and recurrence conditions are needed for convergence from the chosen starting point.

The transition density from $(y,x)$ to $(y^{\prime },x^{\prime })$ isSuppose $(Y,X)$ currently has density $f_{XY}$. The density after one sweep isThis proves the stationarity of the two-coordinate Gibbs sampler:

WriteMultiplying the likelihood by the improper prior gives the Bayesian posterior kernel

Sincethe first full conditional isHolding $\mu$ fixed and comparing powers and exponential rates givesThe assumed parity of $n$ makes $m$ a positive integer.

Initialize ${\mu }_0$ arbitrarily. At iteration $j$, putDraw $m$ independent uniform variables $U_{j1},\dots ,U_{jm}$ and setBy sampling an integer-shape gamma distribution, this has the required $\mathrm{Gamma}(m,r_j)$ law. Next draw $G_j\sim N(0,1)$ independently and setThese are exactly the two full-conditional updates of the normal mean-precision Gibbs sampler. After discarding burn-in, the pairs $({\mu }_j,{\omega }_j)$ are approximate samples from the posterior by Markov chain Monte Carlo.