Question: Let X and Y have a bivariate normal distribution with E(X) = 5, E(Y ) = −2, var(X) = 4,var(Y ) = 9, and cov(X, Y ) = −3. U and V are defined as U = 3X + 4Y and V = 5X − 6Y .Determine the joint distribution of U and V .
Can anyone please provide me some pointers or some similar examples ?
If $A$ is a matrix with $\left(U,V\right)^{T}=A\left(X,Y\right)^{T}$ then $\mathbb{E}\left(U,V\right)^{T}=A\mathbb{E}\left(X,Y\right)^{T}$.
If $\Sigma$ is the covariancematrix of random vector $\left(X,Y\right)^{T}$ then $A\Sigma A^{T}$ is the covariance matrix of random vector $\left(U,V\right)^{T}$.
$\left(U,V\right)^{T}$ has normal distribution, so its distribution is determined by its expectation and covariance matrix.
In your case $A$, $\Sigma$ and $\mathbb E(X,Y)^T$ are given, so everything is "ready" to determine this distribution.