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 $n\to\infty$.