1) Let X₁, X₂, ..... , X_n be a random sample from a population with true mean, µ and true variance, σ². Let the sample mean be estimated as

X = a₁X₁ + a₂X₂ + ....... + a_nX_n,

where a₁, a₂, cdots, an are non-negative constants.

a) What is the condition, a₁, a₂,....., a_n must satisfy so that X is an unbiased estimator of µ?

b) Find the variance of X.

c) What are the conditions, a₁, a₂,....., an must satisfy so that X is a minimum variance unbiased estimator of µ? Just give the condition. You do not have to solve and show when exactly the condition will be met. Your solution will be of the form, some expression is minimum subject to some constraint.

2) Let X₁ be the sample mean of n samples of a normal population with mean, µ and variance σ₂. Let X₂ be the sample mean of n samples of a normal population with mean µ and variance, σ₂. Let the two sets of samples be independent of each other.

a) Show that, ∀ ω ∈ [0, 1] (i.e., 0 ≤ ω ≤ 1), ωX'₁ + (1 - ω)X'₂ is an un-biassed estimator of µ.

b) Show that the variance of this estimator is minimum when

ω = σ₂²/σ₁² + σ₂²

3) A random sample of the performance of 100 students in a test is taken. The true mean, µ was found to be 75 and the true variance, σ₂, was found to be 256.

a) Find the probability that the sample mean, X' falls between 67 and 83.

b) Repeat part #3a) using Chebyshev inequality.

4) A telecommunication network design Engineer wishes to determine the channel capacity of a wireless channel. She performs n = 16 experiments and finds the sample average capacity (based on these 16 experiments) to be 14 Mbps.

a) If the true variance of the channel capacity is 16 (Mbps)2, then what would be the range of the true channel capacity if the Engineer wishes to be 95% confident.

b) if she used the sample variance (which is also 16 (Mbps)2), instead of the true variance.

5) Consider the following data of lengths (in feet) of steel beams rolled in a steel mill.

11.8, 12.1, 12.5, 11.7, 11.9, 12.0, 12.2, 11.5, 11.9, 12.2.

a) Find the error in the claim that the average length of the beam is 12 feet.

b) If you want to make a claim about the mean length of the manufactured steel beams with 99% confidence, what would be the range of the mean?