I have a task:
Random Variable $(X,Y,Z)$ has normal distribution with expected value $EX=0, EY=EZ=1$ and covariance matrix: $$\left[\begin{array}{ccc} 1&1&0\\ 1&2&1\\ 0&1&2\end{array}\right].$$ Calculate $\operatorname{Var}(X(Y+Z))$
From this I know, that $X,Z$ are independent and $$EX^2=1, EXY=1,EY^2=3, EYZ=2,EZ^2=3$$
Simplify $$\operatorname{Var}(X(Y+Z))=E(X(Y+Z))^2-(EX(Y+Z))^2$$ I don't know how calculate $E(X(Y+Z))^2$. Thanks in advance for help.
Note that $$ \Sigma = \left(\begin{array}{ccc} 1&1&0\\ 1&2&1\\ 0&1&2\end{array}\right) = \left(\begin{array}{ccc} 1&0&0\\ 1&1&0\\ 0&1&1\end{array}\right)\,\cdot \, \left(\begin{array}{ccc} 1&0&0\\ 1&1&0\\ 0&1&1\end{array}\right)^T = BB^T, $$ You can transform the vector $(X,Y,Z)^T$ into a vector $(U,V,W)^T$ consisting of independent standard normal random variables by $$ B^{-1}\cdot\pmatrix{X\cr Y-1\cr Z-1} = \pmatrix{U\cr V\cr W} $$ Then $$ \pmatrix{X\cr Y-1\cr Z-1} =B\cdot \pmatrix{U\cr V\cr W} = \pmatrix{U\cr U+V\cr V+W} $$ So we can replace $X=U$, $Y=U+V+1$, $Z=V+W+1$ and therefore $$ X(Y+Z) = U(U+2V+W+2)=U(U+T) $$ where $U$ and $T$ are independent, $U\sim \mathcal N(0,1)$, $T=2V+W+2\sim\mathcal N(2,5)$. So $$ \mathbb E[X^2(Y+Z)^2] = \mathbb E[U^2(U^2+2UT+T^2)] $$ $$= \underbrace{\mathbb E[U^4]}_3+2\underbrace{\mathbb E[U^3]}_0\cdot\mathbb E[T] + \underbrace{\mathbb E[U^2]}_1\cdot\underbrace{\mathbb E[T^2]}_9 = 12. $$