QUESTION $5.$

Given that

$R_1$ and $R_2$ are random variables; $w_1$ and $w_2$ are constants.

Which of the following expression is equal to Var$[{\displaystyle \underset{i=1}{\overset{2}{}}}(w_i{R}_i)]?$

You MUST provide a mathematical proof to justify your answer with the $8$ basic

properties for statistical operators from Lecture #$01.$

A$.,R_1$Var$[w_1]+R_2$Var$[w_2]+2R_1{R}_2$Cov$(w_1,w_2)$

B$.,w_1$Var$[R_1]+w_2$Var$[R_2]+2w_1{w}_2$Cov$(R_1,R_2)$

C$.,R_1^2$Var$[w_1]+R_2^2$Var$[w_2]+2R_1{R}_2$Cov$(w_1,w_2)$

D$.w_1^2$Var$[R_1]+w_2^2$Var$[R_2]+2w_1{w}_2$Cov$(R_1,R_2)$

E$.$ None of the above.

QUESTION $6.$

Let $x$ be the outcome of your first rolling of a dice $($e$.$g$.1,2,3,4,5,6).$

Let $Y$ be the outcome of your second rolling of a dice $($e$.$g$.1,2,3,4,5,6).$

Let's define $Z$ as $(x+Y).$

Calculate the probabilities for various possible outcomes of $Z$ :