Finding bias and variances of estimators of two binomial distributed variables, and my solutions, are they correct?

A product-lot arrives in two containers with respectively 300 and 700 units in each container. We examine 30 units in the first container and find that X1 of them is defective. We check 70 units in the other containter and find that X2 of them is defective.

The following estimators for the defect rate p are proposed:


A) What is the distribution of X1 and X2, and what expectation and variance do they have?

X1 og X2 follows a binomial distribution.

Expected value: E(X1 + X2) = E(X1) + E(X2) = 30p + 70p = 100p

Var(X1 + X2) = Var(X1) + Var(X2) = 30p(1-p) + 70p(1-p) = 100p(1-p)

b) Find the expectation of each of the estimators. Which of them is unbiased?


All look like equally unbiased

c) Find the variance of each of the estimators. Which is best? Try to explain why this one the estimator is better than the others.



The first estimator is best, because it has the lowest variance.

d) Why can the estimators p3 and p4 be less robust than the first two estimators?
How would you interpret it if the two estimators p3 and p4 give very different results?