Answer ALL Six Questions.\ QUESTION 1.\ Given that

,x_(i)=i

,\ Calculate the value of

\\\\sum_(i=1)^n x_(i)

when

(n=5)

.\ QUESTION 2.\ Given that

,x_(i)=i

and

y_(j)=(2\\\\times j)

\ Calculate the value

\\\\sum_(i=1)^n \\\\sum_(j=1)^m {(x_(i)\\\\times y_(j))}

when

(n=3)

and

(m=3)

.\ QUESTION 3.\ Which of the following expression is equal to

\\\\sum_(i=1)^n \\\\sum_(j=1)^m {(w_(i)\\\\times w_(j)\\\\times \\\\sigma _(ij))}

when

(n=2)

and

(m=2)?

\ A.

,(w_(1)\\\\times w_(1)\\\\times \\\\sigma _(11))+(w_(2)\\\\times w_(2)\\\\times \\\\sigma _(22))

\ B.

(w_(1)\\\\times w_(2)\\\\times \\\\sigma _(12))+(w_(2)\\\\times w_(1)\\\\times \\\\sigma _(21))

\ C.

(w_(1)\\\\times w_(1)\\\\times \\\\sigma _(11))+(w_(1)\\\\times w_(2)\\\\times \\\\sigma _(12))+(w_(2)\\\\times w_(1)\\\\times \\\\sigma _(21))+(w_(2)\\\\times w_(2)\\\\times \\\\sigma _(22))

\ D.

(w_(1)\\\\times w_(1)\\\\times \\\\sigma _(11))^(2)+(w_(2)\\\\times w_(2)\\\\times \\\\sigma _(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[\\\\sum_(i=1)^2 (w_(i)\\\\times 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)\\\\times E[w_(1)]+R_(2)\\\\times E[w_(2)]

\ B.

w_(1)\\\\times E[R_(1)]+w_(2)\\\\times E[R_(2)]

\ C.

R_(1)^(2)\\\\times E[w_(1)]+R_(2)^(2)\\\\times E[w_(2)]

\ D.

,w_(1)^(2)\\\\times E[R_(1)]+w_(2)^(2)\\\\times E[R_(2)]

\ E. None of the above.