Question

Let Y₁, Y₂, Yₙ denote a random sample from a population with mean μ and variance 2. Consider the following three estimators for μ:

Answer 1

Complete Question:

Consider three estimators for the population mean μ based on a random sample Y₁, Y₂, Yₙ with mean μ and variance 2. The three estimators are:

1.
`\( \hat{μ}_1 = Y₁ \)`

2.
`\( \hat{μ}_2 = (Y₁ + Y₂)/(2) \)`

3.
`\( \hat{μ}_3 = (Y₁ + 2Y₂ + Yₙ)/(4) \)`

Which of these estimators is unbiased for μ?

Final Answer:

Estimator
`\( \hat{μ}_1 = Y₁ \)` is unbiased for μ.

The correct answer is 1.

Step-by-step explanation:

To determine whether an estimator is unbiased, we need to check if the expected value of the estimator equals the parameter being estimated. In this case, we need to evaluate
`\( E(\hat{μ}_i) \)` for each estimator
`\( \hat{μ}_i \).`

1. For
`\( \hat{μ}_1 = Y₁ \)`:

`\[ E(\hat{μ}_1) = E(Y₁) = μ \]`

Hence,
`\( \hat{μ}_1 \)` is unbiased for μ.

2. For
`\( \hat{μ}_2 = (Y₁ + Y₂)/(2) \)`:

`\[ E(\hat{μ}_2) = (E(Y₁) + E(Y₂))/(2) = μ + (μ)/(2) \\eq μ \] \( \hat{μ}_2 \)`is biased.

3. For
`\( \hat{μ}_3 = (Y₁ + 2Y₂ + Yₙ)/(4) \)`:

`\[ E(\hat{μ}_3) = (E(Y₁) + 2E(Y₂) + E(Yₙ))/(4) = μ + (μ)/(2) + (μ)/(4) \\eq μ \] \( \hat{μ}_3 \)` is biased.

Therefore, only
`\( \hat{μ}_1 \`) is an unbiased estimator for μ.

The correct answer is 1.