Question

Two zero mean Gaussian random variables $X_1$ and $X_2$ have variances respectively; Their covariance is $C_{x_1{x}_2}=3$. These variables are linearly transformed to new random variables as follows:
${\sigma }^2{x}_1=4$ and ${\sigma }^2{x}_2=9$
The value of covarience $C_{Y1Y2}$ will be_____
$Y_1=X_1-2X_2$ and $Y_2=3X_1+4X_2$

• A. -66

Answer 1

The correct option is A -66
Given that, $X_1$ and $X_2$ are zero mean Gaussion random variables,
So, $E\left[X_1\right]={\overline{X}}_1=0andE\left[X_2\right]={\overline{X}}_2=0$

$E\left[X_1^2\right]={\sigma }_{x_1}^2+({\overline{x}}_1{)}^2={\sigma }_{x_1}^2=4$
$E\left[X_2^2\right]={\sigma }_{x_2}^2+({\overline{X}}_2{)}^2={\sigma }_{x_2}^2=9$
$C_{x_1{x}_2}=E\left[X_1{X}_2\right]-E\left({\overline{X}}_1{\overline{X}}_2\right)=E\left[X_1{X}_2\right]=3$
$E\left[Y_1\right]=E\left[X_1\right]-2E\left[X_2\right]=0$
$E\left[Y_2\right]=3E\left[X_1\right]+4E\left[X_2\right]=0$
So, $C_{Y_1{Y}_2}=E\left[Y_1{Y}_2\right]=E\left[\right(X_1-2X_2\left)\right(3X_1+4X_2\left)\right]$
$=E[3X_1^2-8X_2^2-2X_1{X}_2]=3{\sigma }_{X_1}^2-8{\sigma }_{X_2}^2-2E\left[X_1{X}_2\right]$
$=3\left(4\right)-8\left(9\right)-2\left(3\right)=-66$