Let X and Y be independent identically distributed random variables with means µx and µy respectively....

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Let X and Y be independent identically distributed random variables with means µx and µy respectively....

Let X and Y be independent identically distributed random variables with means µ_x and µ_y respectively. Prove the following.

a. E [aX + bY] = aµ_x + bµ_y for any constants a and b.

b. Var[X²] = E[X²] − E[X]²

c. Var [aX] = a²Var [X] for any constant a.

d. Assume for this part only that X and Y are not independent. Then Var [X + Y] = Var[X] + Var[Y] + 2(E [XY] − E [X] E[Y]).

e. Use the previous part and the assumption that X and Y are independent to show that Var [aX + bY ] = a² Var [X] + b² Var [Y ] for any constants a and b.

math Statistics-And-Probability

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Answer 1

Answer #1

for your clarification ragarding proof b:

Var(X2i)=E[(X2i)2]−(E[X2i])2Var(Xi2)=E[(Xi2)2]−(E[Xi2])2
Var(X2i)=E[X4i]−(E[X2i])2Var(Xi2)=E[Xi4]−(E[Xi2])2
E[X4i]E[Xi4] is the 4th moment

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Answer 2

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Let X and Y be independent identically distributed random variables with means µx and µy respectively....

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