Suppose $x_1\sim\mathcal N(2,0.5),x_2\sim \mathcal N(2,3),$ and $x_3\sim \mathcal N(2.5,7)$ with correlations $\rho_{(1,2)}=0.3,\rho_{(1,3)}=0.1,$and $\rho_{(2,3)}=0.4.$ What is the distribution of

1. $x_1+x_2+x_3?$

2. $x_1+ (3\times x_2)+x_3?$

3. $x_1+x_2+(0.5\times x_3)?$

Solutions:

1. The distribution of $x_1+x_2+x_3 =\mathcal{N}(6.5,15.275)$.

2. The distribution of $x_1+(3\times x_2)+x_3=\mathcal{N}(10.5,30.077)$

3. The distribution of $x_1+x_2+(0.5\times x_3)=\mathcal{N}(5.25,8.005)$

But I want to know what is the distribution of $(2\times x_1)-(3\times x_2)-x_3$ and $x_1+x_2-(2\times x_3)?$ How to compute these distributions?

Suppose $x_1\sim\mathcal N(2,0.5),x_2\sim \mathcal N(2,3),$ and $x_3\sim \mathcal N(2.5,7)$ with correlations $\rho_{(1,2)}=0.3,\rho_{(1,3)}=0.1,$and $\rho_{(2,3)}=0.4.$ What is the distribution of

1. $x_1+x_2+x_3?$

2. $x_1+ (3\times x_2)+x_3?$

3. $x_1+x_2+(0.5\times x_3)?$

Solutions:

1. The distribution of $x_1+x_2+x_3 =\mathcal{N}(6.5,15.275)$.

2. The distribution of $x_1+(3\times x_2)+x_3=\mathcal{N}(10.5,30.077)$

3. The distribution of $x_1+x_2+(0.5\times x_3)=\mathcal{N}(5.25,8.005)$

But I want to know what is the distribution of $(2\times x_1)-(3\times x_2)-x_3$ and $x_1+x_2-(2\times x_3)?$

The distribution of $(2\times x_1)-(3\times x_2)-x_3$ is $\mathcal{N}(-4.5,41.8407).$

The distribution of $x_1 +x_2-(2\times x_3) $ is $\mathcal{N}(-1,24.1544)$

Suppose $x_1\sim\mathcal N(2,0.5),x_2\sim \mathcal N(2,3),$ and $x_3\sim \mathcal N(2.5,7)$ with correalations $\rho_{(1,2)}=0.3,\rho_{(1,3)}=0.1,$and $\rho_{(2,3)}=0.4.$ What is the distribution of

1)$x_1+x_2+x_3?$

2)$x_1+ (3\times x_2)+x_3?$

3)$x_1+x_2+(0.5\times x_3)?$

Solutions

1. The distribution of $x_1+x_2+x_3 =\mathcal{N}(6.5,15.275)$.

2)The distribution of $x_1+(3\times x_2)+x_3=\mathcal{N}(10.5,30.077)$

3)The distribution of $x_1+x_2+(0.5\times x_3)=\mathcal{N}(5.25,8.005)$

But i want to know what is the distribution of $(2\times x_1)-(3\times x_2)-x_3$ and $x_1+x_2-(2\times x_3)?$ How to compute these distributions?

Suppose $x_1\sim\mathcal N(2,0.5),x_2\sim \mathcal N(2,3),$ and $x_3\sim \mathcal N(2.5,7)$ with correlations $\rho_{(1,2)}=0.3,\rho_{(1,3)}=0.1,$and $\rho_{(2,3)}=0.4.$ What is the distribution of

1. $x_1+x_2+x_3?$

2. $x_1+ (3\times x_2)+x_3?$

3. $x_1+x_2+(0.5\times x_3)?$

Solutions:

1. The distribution of $x_1+x_2+x_3 =\mathcal{N}(6.5,15.275)$.

2. The distribution of $x_1+(3\times x_2)+x_3=\mathcal{N}(10.5,30.077)$

3. The distribution of $x_1+x_2+(0.5\times x_3)=\mathcal{N}(5.25,8.005)$

But I want to know what is the distribution of $(2\times x_1)-(3\times x_2)-x_3$ and $x_1+x_2-(2\times x_3)?$ How to compute these distributions?