Any help on either/both of these would be greatly appreciated.

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Suppose that P(Y=i)=p(i) for i=1,2,...,m. Assume that you can simulate X(i) which corresponds to the value of X given that Y=i. We are interested in estimating =E(X) using optimal stratified sampling. Consider the estimator

(n1,...,nm)=i=1mnip(i)j=1niXj(i).

where the Xj(i)'s are i.i.d. copies of X(i) and all of the random variables involved are independent.

1. Use calculus to find an expression for the values (n1,...,nm) that minimizes the variance of (n1,...,nm) subject to the constraint that n1++nm=N. Feel free to treat the ni's as continuous variables for this exercise.
2. Discuss briefly how to implement an estimator based on the optimal ni's given that such estimator depends on Var(X(i)), which is unknown.