density of 2 bivariate gaussian random variables

Question

$X_1$ and $X_2$ are bi-variate Gaussian with equal mean and variance. how do i find the density of & $y = A_1X_1 + B_1X_2$.? I think I should use correlation co-efficient here which i assume as $p$. but cant find a way really. any suggestion plz.

Answer 1

To specify a bivariate normal distribution, one can specify the two means, the two variances, and the covariance. You haven't said anything about the covariance. You would have $$ \operatorname{var}(A_1X_1+B_1X_2) = A_1^2\operatorname{var}(X_1) + B_1^2\operatorname{var}(X_2) + 2A_1B_2\operatorname{cov}(X_1,X_2). $$ The expected value of $A_1X_1+B_1X_2$ is easier to find than that. Once you've to the variance and expected value, do you know how to go from there to the density?