Monte Carlo Mean Estimation

Question

Monte Carlo: to find out the mean of $A$, find a variable $B$, $corr(A,B)=c$, and simulate $A + B(E(B)-B)$ instead of A. What is $B$? The choices of B are cov(a,b)/var(a), cov(a,b)/var(b), 1, -1

Answer 1

$$corr(A,B)=\frac{E(A-E(A))E(B-E(B))}{\sigma_A\sigma_B}=c$$ $${E(B-E(B))}=\frac{c(\sigma_A\sigma_B)}{E(A-E(A))}$$ $$A+B({E(B)-B)}=A+\bigg(\frac{c(\sigma_A\sigma_B)}{E(A-E(A))}-B\bigg)B$$