Applied Stats: Distributed Averaging Under Noisy Communication

Question

# Distributed Averaging Under Noisy Communication

You are given a fully connected network of N nodes. Node i holds a real-valued measurement Xi.  A designated aggregator node (node 1) must compute the average value:

μ = (1 / N) · Σ Xi    (sum over i = 1 … N)

Communication between nodes may be noisy.

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## Part 1 — Noiseless Baseline

Assume communication is perfect: when a node sends a value, the receiver gets it exactly.

1. Describe a simple protocol that allows node 1 to compute μ exactly.

2. State the total number of messages required as a function of N.

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## Part 2 — Noisy Communication and Statistical Estimation

Now assume communication is corrupted by additive Gaussian noise.

When node i sends a real value v, the receiver observes:

Y = v + ε

where ε is independent noise distributed as:

ε ~ Normal(0, σ²)

Node 1 wants to estimate the value Xi held by node i.

Node i sends Xi exactly k times, producing observations:

Y₁, Y₂, …, Yₖ
    with Yₜ = Xi + εₜ

Node 1 estimates Xi using the sample mean:

Ȳ = (1 / k) · Σ Yₜ

Answer the following:

1.Derive the distribution of Ȳ (its mean and variance).

2.Given an absolute error tolerance τ > 0 and c

Accepted Answer

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