Variance, Covariance, and Correlation answer check

Question

Two random variables, $Y$ and $Z$:

$Y = 0.5+0.6X$

$Z = 0.2+0.3X$

where $X$ is another random variable. You can treat the variance $var(X)$ as a given constant. It may help to give $var(X)$ a name, $()=^2$.

$var(Y) = 1/2 + (9/25)^2$

$var(Z) = 2/25 + (9/100)^2$

$Cov(Y,Z) = (9/50)^2$

$Cov(X,Z) = (3/10)^2$

$Cov(X,Y) = (3/5)^2$

Also, how would I find the $Corr(Y,Z)$, $Corr(X,Z)$, and $Corr(X,Y)$?

Answer 1

Firstly. If $Y=0.5+0.6X$ and $\mathsf {Var}(X)=\sigma^2$ , then $\mathsf{Var}(Y)=0.36\sigma^2$

Because $\mathsf {Var}(a+bX) ~=~ b^2~\mathsf{Var}(X)$ when $a,b$ are constants.

Similarly: $\mathsf {Cov}(a+bX, c+dX) ~=~ bd~\mathsf{Var}(X)$

Revisit all your calculations.

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Secondly The correlation coefficient is defined as:

$$\mathsf {Corr}(U,V) ~=~ \dfrac{\mathsf {Cov}(U,V)}{\sqrt{~\mathsf{Var}(U)~\mathsf{Var}(V)~}}$$

Just substitute as appropriate.