Hello, my question is related to the score statistic of a binomial distribution.

The book does it by finding var(U), subbing in U, and then solving with the knowledge that Y =n*pi*(1-pi), which I understand.

I tried another method, using var(U) = E(U^2). To do this I subbed in U^2 and solved. This did not give me the same answer and I have tried multiple times. Can I not do it this way?

Edit: I'm now looking directly at the score function:

$$ U^2=a(y)^2b'(theta)^2+c'(theta)^2+2a(y)b'(theta)c'(theta) $$

Thus:

$$ E[U^2] = E[a(y)^2]b'(theta)^2+c'(theta)^2+2E[a(y)] b'(theta)c'(theta) $$

Subbing in E[a(Y)]:

$$ = E[a(y)^2]b'(theta)^2-c'(theta)^2 $$

Which is var(U) + an extra term. What is wrong with this calculation?

Edit2: Nevermind, I went wrong in the calc :)