Combine noisy thermometers; compute random-walk correlations

## Problem 1: Estimating a true temperature from noisy thermometers Assume the true (fixed) temperature is an unknown constant \(\theta\). ### 1a) One thermometer You take \(n\) independent measurements from thermometer A: \[X_1,\dots,X_n\] with - \(\mathbb E[X_i]=\theta\) (unbiased for the true temperature), - \(\mathrm{Var}(X_i)=\sigma_1^2\), - measurements are i.i.d. **Task:** Propose an estimator \(\hat\theta\) of \(\theta\). What is its variance? ### 1b) Two thermometers You also take \(m\) independent measurements from thermometer B: \[Y_1,\dots,Y_m\] with - \(\mathbb E[Y_j]=\theta\), - \(\mathrm{Var}(Y_j)=\sigma_2^2\), - all \(X\)’s are independent of all \(Y\)’s. **Task:** Construct a combined estimator \(\hat\theta\) using both thermometers that minimizes variance among **linear unbiased** estimators. Give the optimal weights and the resulting variance. > If you need an additional assumption, you may assume \(\sigma_1^2\) and \(\sigma_2^2\) are known. --- ## Problem 2: Correlations in a 2D random walk Define a 2D simple random walk starting at \((0,0)\). At each step you move **one unit** in exactly one of the four directions \(\{\text{up,down,left,right}\}\), each with probability \(1/4\). Let \((X_n,Y_n)\) be the coordinates after \(n\) steps. ### 2a) Compute \(\mathrm{Corr}(X_n, Y_n)\). ### 2b) Compute or characterize \(\mathrm{Corr}(|X_n|, |Y_n|)\). If a simple closed form is hard, give a correct expression and determine at least the **sign** and **asymptotic behavior** as \(n\to\infty\).