Is W an Unbiased Estimator of Mu?

[joemama69]

Homework Statement

Let Y_1, Y_2, Y_3, Y_4 be IID RV from a population with mean mu and variance sigma^2. Let Y^bar = .25(Y_1+Y_2+Y_3+Y_4) denote the average of these four RV's.

1)What are the expected value and variance of Y^bar in terms of mu and sigma^2

2)Consider a different estimator of mu... W = (1/8)Y_1 + (1/8)Y_2 + (1/4)Y_3 + (1/2)Y_4
This is an example of a weighted average of Y_i. Show that W is also an unbiased estimator of mu and find V(W)

3)Based on your answer, which estimator of mu do you prefer, Y^bar or W.

Homework Equations

The Attempt at a Solution

Welp I'm not really sure where to start. I know that Y^bar is a sample mean from 4 samples while mu is the population mean based on every sample, but I do not know how to represent one in terms of the other

 

[LCKurtz]

Homework Statement

Let Y_1, Y_2, Y_3, Y_4 be IID RV from a population with mean mu and variance sigma^2. Let Y^bar = .25(Y_1+Y_2+Y_3+Y_4) denote the average of these four RV's.

1)What are the expected value and variance of Y^bar in terms of mu and sigma^2

2)Consider a different estimator of mu... W = (1/8)Y_1 + (1/8)Y_2 + (1/4)Y_3 + (1/2)Y_4
This is an example of a weighted average of Y_i. Show that W is also an unbiased estimator of mu and find V(W)

3)Based on your answer, which estimator of mu do you prefer, Y^bar or W.

Homework Equations

The Attempt at a Solution

Welp I'm not really sure where to start. I know that Y^bar is a sample mean from 4 samples while mu is the population mean based on every sample, but I do not know how to represent one in terms of the other

Doesn't your text have formulas for the expected value and variance of a sum (linear combination) of IID random variables?

 

[joemama69]

Doesn't your text have formulas for the expected value and variance of a sum (linear combination) of IID random variables?

Sure does, but how to I express mean and variance IN TERMS of mu and sigma^2

Heres what I got outa the book...

so Y^bar = Sum(Y_i/n)

E(Y^bar) = E(Sum(Y_i/n)) = (1/4)Sum(E(Y_i)) = (1/4)Sum(mu)

V(Y^bar) = V(Sum(Y_i/n)) = (1/16)V(Sum(Y_i)) = (1/16)(Sum(Y_i) + 2 SumSumcov(Y_i, Y_j)) = (1/16)Sum(V(Y_i)) = (1/16)Sum(sigma^2)

So that's part one. Part two I am lossed with the weighted average parts. How would I go about that

 

[LCKurtz]

Sure does, but how to I express mean and variance IN TERMS of mu and sigma^2

Heres what I got outa the book...

so Y^bar = Sum(Y_i/n)

E(Y^bar) = E(Sum(Y_i/n)) = (1/4)Sum(E(Y_i)) = (1/4)Sum(mu)

Hopefully you are using proper notation on your worksheet. When you write (1/4)sum(mu) what you really mean is$$
\frac 1 4\sum_{i=1}^4\mu$$What does that equal when you write it out?

V(Y^bar) = V(Sum(Y_i/n)) = (1/16)V(Sum(Y_i)) = (1/16)(Sum(Y_i) + 2 SumSumcov(Y_i, Y_j)) = (1/16)Sum(V(Y_i)) = (1/16)Sum(sigma^2)

So that's part one. Part two I am lossed with the weighted average parts. How would I go about that

Same comment about the variance sums. And do you need the covariance for IID random variables?

 

[joemama69]

Ya sorry about the notation, I'm not up to speed with this syntax

So now my question is regarding part two... How would I find E(W) and V(W) when W = the sum of the weighted averages of Y (as apposed to part 1 where Y^bar was just the sum of the Y's divided by n = 4). Basically I'm confused what to do with the weights i.e. 1/8, 1/8, 1/4, 1/2?

 

[LCKurtz]

Ya sorry about the notation, I'm not up to speed with this syntax

So now my question is regarding part two... How would I find E(W) and V(W) when W = the sum of the weighted averages of Y (as apposed to part 1 where Y^bar was just the sum of the Y's divided by n = 4). Basically I'm confused what to do with the weights i.e. 1/8, 1/8, 1/4, 1/2?

You have $W=\sum c_iY_i$ with $Y_i$ IID. Look in your text for the mean and variance formulas for such sums.