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).