Answer ALL Six Questions.

QUESTION $1.$

Given that $,x_i=i,$

Calculate the value of ${\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i$ when $(n=5).$

QUESTION $2.$

Given that $,x_i=$i and $y_j=(2j)$

Calculate the value ${\displaystyle \underset{i=1}{\overset{n}{}}}{\displaystyle \underset{j=1}{\overset{m}{}}}\{(x_i{y}_j)\}$ when $(n=3)$ and $(m=3).$

QUESTION $3.$

Which of the following expression is equal to ${\displaystyle \underset{i=1}{\overset{n}{}}}{\displaystyle \underset{j=1}{\overset{m}{}}}\{(w_i{w}_j{}_{ij})\}$ when $(n=2)$ and $(m=2)?$

A$.,(w_1{w}_1{}_{11})+(w_2{w}_2{}_{22})$

B$.(w_1{w}_2{}_{12})+(w_2{w}_1{}_{21})$

C$.(w_1{w}_1{}_{11})+(w_1{w}_2{}_{12})+(w_2{w}_1{}_{21})+(w_2{w}_2{}_{22})$

D$.(w_1{w}_1{}_{11}{)}^2+(w_2{w}_2{}_{22}{)}^2$

E$.$ None of the above.

QUESTION $4.$

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 $E[{\displaystyle \underset{i=1}{\overset{2}{}}}(w_i{R}_i)]?$

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

properties for statistical operators from Lecture #$01.$

A$.,R_{1}E[w_1]+R_{2}E[w_2]$

B$.w_{1}E[R_1]+w_{2}E[R_2]$

C$.R_1^{2}E[w_1]+R_2^{2}E[w_2]$

D$.,w_1^{2}E[R_1]+w_2^{2}E[R_2]$

E$.$ None of the above.