 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 216

By the <stationary distribution> property, $X_1$ and $X_2$ have the same <marginal distribution> $\pi$. Expanding the square gives
$$
\begin{aligned}
\frac12\mathbb E[(f(X_2)-f(X_1))^2]
&=\frac12\left(2\langle f,f\rangle_\pi
-2\mathbb E[f(X_1)f(X_2)]\right)\\
&=\langle f,f\rangle_\pi-\langle f,Kf\rangle_\pi\\
&=\langle f,(I-K)f\rangle_\pi
=\mathcal E_K(f).
\end{aligned}
$$
This is the probabilistic representation of the <Dirichlet form of a Markov chain>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 3 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 3 / c / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 3 / c / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)