Let X₁, X₂ be two normal random variables each with population mean and population variance ². Let ₁₂ denote the covariance between X₁ and X₂ and let X denote the sample mean of X₁ and X₂.

1) List the condition that needs to be satisfied in order for X to be an unbiased estimate of

2) As carefully as you can, without skipping steps, show that both X₁ and X are unbiased estimators of

3) Assuming X₁ and X₂ are independent to each other (₁₂ = 0), as carefully as you can, without skipping steps, show that X is more efficient than X₁

4) Assuming X₁ and X₂ are dependent to each other with cov(X₁,X₂) = ₁₂, as carefully as you can, without skipping steps, show that Var(X) = ² + ₁₂) / 2

5) Let = 5, = 10, and ₁₂ = 10, find P(-1 X 1)

6) Instead of normally distributed random variables, Let X₁, X₂... X₁₀₀₀ be one thousand independently uniformly distributed random variables with the same closed interval {0,5}. Find the density function for X₁, i = 1,2,....,1000

7) Refer to [6], what is the probability that X₁ is greater than 2?

8) Refer to [6], let X denote the sample mean of X₁, X₂... X₁₀₀₀. what is the E(X) now?

9) Refer to [6], assume you know the population variance for X_i, Var(X_i) = 25/3, what is the probability that X is less or equal to 2? Carefully specify the theorem used to determine the probability