1. Let Y ₁ , Y ₂ , Y ₃ , and Y ₄ be independent random variables with means  ₁ ,  ₂ ,  ₃ ,  ₄ , and equal variance  ² . For each of the following linear combinations of these Y’s determine the coefficients c ₁ , c ₂ , c ₃ , c ₄ in the general formula Z = c ₁ Y ₁ + c ₂ Y ₂ + c ₃ Y ₃ + c ₄ Y ₄ .

1. Z ₁ = Y ₁ + Y ₂ .
2. Z ₂ = Y ₁ - Y ₂ .
3. Z ₃ = (Y ₁ + Y ₂ )/2 - (Y ₃ + Y ₄ )/2
4. Z ₄ = (Y ₁ - Y ₂ )/2 - (Y ₃ - Y ₄ )/2
5. Z ₅ = Y ₁ - (Y ₂ + Y ₃ + Y ₄ )/3

1. In the setting of problem 1, find E(Z _j ) for each Z _j , j =1,2,3,4,5 in terms of  ₁ ,  ₂ ,  ₃ ,  ₄ .
2. In the setting of problem 1, find Var(Z _j ) for each Z _j , j=1,2,3,4,5 as a multiple of  ² .
3. If S ² is the sample variance for a random sample of size n from a N(, ² ) population,

derive the expected value and variance of S ² .

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