I am working on a large problem that is broken down into steps. The goal is to find the 95% confidence interval for p(x=1).

X takes on the value 1 if unemployed, 0 otherwise.

Y takes on the value 1 if no response, 0 otherwise.

I have theta = pr(X=1), theta1= pr(X=1, Y=0) and theta2 = pr(Y=1).

I need to find estimators for theta 1 and theta 2 and prove they are unbiased given the following information.

We sent out n questionnaires. We received n1 back, and of that n1, m are unemployed.

My estimators are:

theta1hat = m/n1

theta2hat = (n-n1)/n

I am confused on how to show they are unbiased. I know to take the expectation, but don't know how to actually do that.

Thanks in advance.