# Normal bivariate distribution.

Suppose that $X_1$ and $X_2$ randon variables are independent with normal distribution $(0,1)$ . Let $Y_1=X_1+3X_2-2$ and $Y_2=X_1-2X_2+1$. Determine the distribution of $Y=(Y_1,Y_2)$

I know that

• $Var(Y_1)=Var(X_1+3X_2-2)=Var(X_1)+9Var(X_2)+6Cov(X_1,X_2)=1+9+0=10$
• $Var(Y_2)=Var(X_1-2X_2+1)=Var(X_1)+4Var(X_2)-4Cov(X_1,X_2)=1+4+0=5$
• And that $\mu=(-2,1)$

Then $\Sigma$=\begin{bmatrix} 10 & 0 \\0 & 5\end{bmatrix}

Since $Cov(Y_1,Y_1)=0$, but the answer in the book is $\Sigma$=\begin{bmatrix} 10 & -5 \\-5 & 5\end{bmatrix}

Am i doing something wrong with the covariance between the variables?

• You're right, i was doing the covariance of $Y_1$ and $Y_2$ wrong. Thank you! – Emma Wool May 24 '17 at 14:56