explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-:...

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explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-: μί ald Var(Xi)-σ? where i-| , 2, , n. If ai, aam., an are constants. (i) Write down expression for (i) E{E:-aiX.) and (ii) Var(Σ-lai%). (i) Rewrite the expression if X,s are not independent.

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Answer 1

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Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-: μί ald Var(Xi)-σ? where i-| , 2, , n. If ai, aam., an are constants. E{E:-aiX.) and (ii) Var(Σ-lai%). The expression if X,s are not independent.

diMi に! に! に!

Var( 〉 aix, a, a,Cov(X., Xi i,j-1 i,j-1

X₁ , X₂ , .... , X_n are independent random variables Rightarrow Cov(X_i , X_j) = 0 i,j=1,2,...,n ; ij

aVar(X)+0 i=1 i=1 i=1

(ii)

Var( 〉 aix, a, a,Cov(X., Xi i,j-1 i,j-1

X₁ , X₂ , .... , X_n are not independent random variables.

Let _ij be the correlation between X_i and X_j i,j=1,2,...,n ; ij

herefore Cov(X_i , X_j) = _ij sigma _i sigma _j i,j=1,2,...,n ; ij

i,j-1 i,j-1

diMi に! に! に!

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