I have a complicated problem in which I have to compute the expectation in order to see if my estimator is unbiased or not. After a lot of calcutions at the end I have found: $$ E[(\sqrt{A-c})(\sqrt{A+c})] $$where A is an expression which contains the random variable and c in a constant.
The exact expression for A and c are the following: $$ A=w(\sum_{i=1}^{g}y_{i}d_{i}-\beta_{0}\sum_{i}^{g}d_{i})) $$ and $$ c=g\beta_{0} $$ where the random variables are $y_{i}$ i.i.d with normal distribution N($\beta_{0}+\beta_{1}d_i$, $\sigma^{2}$) . And this calculation is to prove that $\beta_1$ is unbiased.
I have tryed in a lot of way but I don't really get which trick I should use.. I hope that someone could help me.
Thank you in advance!