I have three standard normal variables, $X_1$, $X_2$, and $X_3 \sim N(0,1)$.
Let $Y_1 = X_1 - X_2$, $Y_2 = X_3 - \bar X_2$, $Y_3 = \bar X_3$
Where $\bar X_2 = (X_1 + X_2)/2$ and $\bar X_3$ is found in the same way.
Show that $Y_1$, $Y_2$, $Y_3$ have a multivariate normal distribution, find $\mu$ and $\Sigma$.
Can someone please guide me on this or point me to a textbook which illustrates these types of transformations?
Edit: Thanks for editing my math, Michael.