Final answer:
The estimator μ = ∑ⁿᵢ₊₁ Xᵢ / n+2 is biased because its expected value does not equal the true population mean. The bias of the estimator is B(μ) = μ(-2/(n+2)).
Step-by-step explanation:
The question asks whether the estimator μ = ∑ⁿᵢ₊₁ Xᵢ / n+2 is an unbiased estimator of the population mean, and if not, to find its bias. To assess if μ is unbiased, we analyze the expected value of the estimator. For an estimator to be unbiased, E(μ) must equal the true population mean (μ).
From the Central Limit Theorem, we know that the distribution of the sample mean is approximately normal with mean E(X) = μ and standard deviation σ/sqrt(n) as the sample size n becomes large. In this scenario, the expected value of the sample mean is μ because E(∑X) = nE(X) = nμ by the properties of expected value. However, the given estimator adds 2 to the denominator, which affects the expectancy.
To calculate the expected value of the given estimator and the bias:
- E(μ) = E(∑ⁿᵢ₊₁ Xᵢ / n+2) = E(∑X)/(n+2)
- E(∑X) = nμ (since E(X) = μ)
- E(μ) = nμ / (n+2)
- B(μ) = E(μ) - μ = (nμ / (n+2)) - μ = μ(n/(n+2) - 1) = μ(-2/(n+2))
Therefore, the estimator is biased, and the bias is B(μ) = μ(-2/(n+2)).