Multivariable Expectation

Question

$E(X)=x_1, E(Y)=y_1$

$Var(X)=x_2, Var(Y)=y_2$

correlation coefficient between X and Y $\rho(X,Y)=C$

need to find $E(Z)$ and $Var(Z)$ where $Z=aX+bY$

Answer 1

Use the linearity property of expectation value $$E[aX+ bY]=aE[X]+bE[Y]$$ and use definition $$\textit{var}[aX+bY]=E[(aX+bY-E[aX+bY])^2]$$

Answer 2

We have that $\operatorname{Var}Z=\operatorname{cov}(Z,Z)$. Also, the covariance is linear in the sense that $$ \operatorname{cov}(aX+bY, cW+dV) = ac\,\operatorname{cov}(X,W)+ad\,\operatorname{cov}(X,V)+bc\,\operatorname{cov}(Y,W)+bd\,\operatorname{cov}(Y,V) $$ for any random variables $X,Y,W,V$ and any real numbers $a,b,c,d$. Hence, $$ \operatorname{cov}(aX+bY,aX+bY)=a^2\operatorname{Var}X+ab\operatorname{cov}(X,Y)+ba\operatorname{cov}(Y,X)+b^2\operatorname{Var}Y. $$ The correlation between $X$ and $Y$ is given by $$ \operatorname{\rho}(X,Y)=\frac{\operatorname{cov}(X,Y)}{\sqrt{\operatorname{Var}X}\sqrt{\operatorname{Var}Y}}. $$ Now we only need to use the given values to calculate $\operatorname{Var}Z$.