Combine noisy thermometers; compute random-walk correlations

Problem 1: Inferring a true temperature from noisy thermometers

Assume the actual (fixed) temperature is an unknown constant $\theta$.

1a) Single thermometer

You collect $n$ independent readings from thermometer A: $X_1,\dots,X_n$ with

• $\mathbb E[X_i]=\theta$ (unbiased for the true temperature),
• $\mathrm{Var}(X_i)=\sigma_1^2$,
• the measurements are i.i.d. Task: Suggest an estimator $\hat\theta$ for $\theta$. What is its variance?

1b) Two thermometers

You also collect $m$ independent readings from thermometer B: $Y_1,\dots,Y_m$ with

• $\mathbb E[Y_j]=\theta$,
• $\mathrm{Var}(Y_j)=\sigma_2^2$,
• all $X$ measurements are independent of all $Y$ measurements. Task: Build a combined estimator $\hat\theta$ from both thermometers that has minimum variance among linear unbiased estimators. State the optimal weights and the resulting variance.

If an extra assumption is needed, you may take $\sigma_1^2$ and $\sigma_2^2$ to be known.

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Problem 2: Correlations in a two-dimensional random walk

Consider a 2D simple random walk that begins at $(0,0)$. On each step, you move one unit in exactly one of the four directions $\{\text{up, down, left, right}\}$, each with probability $1/4$. Let $(X_n,Y_n)$ denote the coordinates after $n$ steps.

2a)

Compute $\mathrm{Corr}(X_n, Y_n)$.

2b)

Compute or describe $\mathrm{Corr}(|X_n|, |Y_n|)$. If a simple closed form is difficult, provide a correct expression and at least identify the sign and asymptotic behavior as $n\to\infty$. Overview: This problem assesses estimation and probabilistic reasoning, particularly building unbiased estimators and variance-minimizing linear combinations to merge noisy measurements, as well as computing and characterizing the asymptotic correlations in a 2D simple random walk.