Streaming Mean and Binomial vs Poisson Approximation

Part A — Streaming Mean Update

You have an existing dataset of N = 1,000 observations with mean 12.4. You:

• Remove r = 50 observations whose mean is 20.1, then
• Append a = 30 new observations whose mean is 9.8.

Without recomputing from raw data, compute the exact new mean. Show the formula you use and report the final value to 3 decimal places.

Part B — Binomial vs Poisson Approximation

Let X ~ Binomial(n = 40, p = 0.07).

1. Compute P(X = 0), P(X ≤ 2), and P(X ≥ 3) exactly. Report each to 4 decimal places.
2. Approximate the same three quantities using a Poisson(λ = n·p) model (use continuity only if appropriate), and report absolute errors for each quantity (|exact − approx|).
3. Briefly justify when the Poisson approximation is appropriate here, and how you would tighten the approximation if p were larger.