This is a more general version of Problem C.1. Let Y1, Y2, …, Yn be n pairwise uncorrelated random variables with common mean m and common variance 2. Let Y denote the sample average. (i) Define the class of linear estimators of m by Wa a1Y1 a2Y2 … anYn,
where the ai are constants. What restriction on the ai is needed for Wa to be an unbiased estimator of m?
(ii) Find Var(Wa).
(iii) For any numbers a1, a2, …, an, the following inequality holds: (a1 a2 … an)2/n a1 a2 … an. Use this, along with parts (i) and (ii), to show that
2 2 2
Var(Wa) Var(Y) whenever Wa is unbiased, so that Y is the best linear unbiased estimator. [Hint: What does the inequality become when the ai satisfy the restriction from part (i)]

$527.7.if the following was the question  $527.7 is the answer.students who are trying to raise funds have an agreement with a local pizza chain. the chain has agreed to sell them pizzas at a discount, which the students can then resell to families in the local community for a profit. it is expected that of the 500 families in the community, at most 70% will buy pizza. based on a survey of their personal preferences, the students believe that they should order no more than 120 cheese pizzas, no more than 150 pepperoni pizzas, and no more than 100 vegetarian pizzas. they also want to make sure that at least 20% of the total pizzas are cheese and at least 50% of the pizzas are pepperoni. they make a profit of $1.45, $1.75, and $1.98, respectively, for each cheese, pepperoni, and vegetarian pizza they resell. how many pizzas of each type should they buy?