Find the variance of W when given $W = x + 2y + 3z$.

Question

1. x,y,z are random numbers given w = x + 2y + 3z. also given that the mean of x,y,z= 1,8,0 respectively. what is the mean of the random number w ?

2. Assuming the Standard deviation of the random numbers x,y,z = 1 and the numbers are uncorrelated, what will the variance of the random number w ?

3. Assuming the Standard deviation of the random numbers x,y,z = 1 and the numbers are uncorrelated except of x,y that correlation of Rxy = 9 what will the variance of the random number w ?

I dont know how to solve this problem, and how to approach it.

Can i get a full solution with explanation please ?

Thanks

Answer 1

For all random variables $U,V$ and constants $a,b$:

Expectation is Linear : $$\mathsf E(aU+bV) ~=~ a\,\mathsf E(U)+b\,\mathsf E(V)$$

Variance is not linear, but we have the rule that : $$\mathsf {Var}(aU+bV) ~=~ a^2\,\mathsf {Var}(U)+b^2\,\mathsf {Var}(V)-2ab\,\mathsf {Cov}(U,V)$$

However, when $U,V$ are uncorrelated : $$\mathsf {Var}(aU+bV)~=~ a^2\,\mathsf {Var}(U)+b^2\,\mathsf {Var}(V)\color{silver}{\require{cancel}\cancelto{0}{-2ab\,\mathsf {Cov}(U,V)}}$$

The correlation cooefficient s $$R_{U,V} ~=~ \dfrac{\mathsf {Cov}(U,V)}{\sqrt{~\mathsf {Var}(U)~\mathsf{Var}(V)~}}$$

Use these rules.