Let $X$ and $Y$ be distributed as bivariate normal random variables with pdf
$$f_{X, Y}(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\biggl(\frac{-1}{2(1-\rho^2)}(x^2-2\rho(xy)+y^2)\biggr)$$
Find the distribution of $aX+bY+c$.
Would this be done by finding the marginal distributions of both $X$ and $Y$ and then working with those to find the desired distribution?
I worked through the problem and used the fact that X varies N(0,1) and Y varies N(0,1).
I then considered $$Var(aX+bY+c)=a^2+b^2+2ab*Cov(X,Y)$$
$$=a^2+b^2+2ab\rho\sigma_x\sigma_y$$
$$=a^2+b^2+2ab\rho$$
Is this it? Then I could say $aX+bY+c$ varies $N(c, a^2+b^2+2ab\rho)$? Seems a little off to me.
Best Answer
That is right. There is a more general way to calculate such affine transformations you probably will see later. We have already established that $$ \begin{pmatrix} X\\ Y \end{pmatrix} \sim N_2\left( \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 1& \rho\\ \rho&1 \end{pmatrix}\right) $$ then $aX+bY+c = (a,b)(X,Y)^T + c$ and we find $$ aX+bY+c \sim N\left( c + (a,b) \begin{pmatrix} 0 \\ 0 \end{pmatrix}, (a,b) \begin{pmatrix} 1& \rho\\ \rho&1 \end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix} \right) = N(c, a^2+b^2 +2ab\rho). $$ This is from a general calculation of affine transformation $Y = \eta + BX$, where $X\sim N_p(\mu,\Sigma)$ is p-dimensional normal distributed, $B$ is a $k\times p$ matrix and $\eta\in \mathbb{R}^k$. Then $$Y \sim N_k(\eta+B\mu, B\, \Sigma \,B^T). $$