Problem 4. This is an exercise about properties of expectations and variances, which we will use extensively in this course. It is closely related to SW Key Concept 2.3. Suppose you only know the following facts of variances and covariances: If X, Y, and Z are any random variables, and a and b are any constants, then I Var (X+Y) Var (aX) Cov (aX, bY) Cov (X+Y,Z) Cov(a, X) = = = = = Var (X) + Var (Y) + 2Cov (X, Y) a²Var (X) abCov (X,Y) Cov (X, Z) + Cov (Y,Z) 0. Also, you know that if X and Y are independent (I will denote that by X LY) then Cov (X, Y) = 0. For calculations below, please provide precise justification for every step / equality in terms of the given information, and avoid intuitive generalizations. (a) Use the above facts to show another related fact: If Z₁ and Z2 are random variables, and C1 and C2 are constants, then Var (c₁Z₁+c₂Z₂) = cVar (Z₁)+cVar (Z₂)+c1c22Cov (Z₁, Z₂). (b) Use any of the above facts to show that if Z₁ and Z₂ are independent, Var (Z₁ + Z2) = Var (Z₁) + Var (Z₂). (c) Use the result in (b) repeatedly to show that if Z₁,..., Z4 are independent, then Var (Z₁ + Z2 + Z3+ Z4) = Var (Z₁) + Var (Z₂) + Var (Z3) + Var (Z₁). These methods can be applied to obtain even more general results, such as Var (1Z₁) = Var (Z₁) or Var (Σ1CZ) = Σ₁c²Var (Z₁). i=1
Problem 4. This is an exercise about properties of expectations and variances, which we will use extensively in this course. It is closely related to SW Key Concept 2.3. Suppose you only know the following facts of variances and covariances: If X, Y, and Z are any random variables, and a and b are any constants, then I Var (X+Y) Var (aX) Cov (aX, bY) Cov (X+Y,Z) Cov(a, X) = = = = = Var (X) + Var (Y) + 2Cov (X, Y) a²Var (X) abCov (X,Y) Cov (X, Z) + Cov (Y,Z) 0. Also, you know that if X and Y are independent (I will denote that by X LY) then Cov (X, Y) = 0. For calculations below, please provide precise justification for every step / equality in terms of the given information, and avoid intuitive generalizations. (a) Use the above facts to show another related fact: If Z₁ and Z2 are random variables, and C1 and C2 are constants, then Var (c₁Z₁+c₂Z₂) = cVar (Z₁)+cVar (Z₂)+c1c22Cov (Z₁, Z₂). (b) Use any of the above facts to show that if Z₁ and Z₂ are independent, Var (Z₁ + Z2) = Var (Z₁) + Var (Z₂). (c) Use the result in (b) repeatedly to show that if Z₁,..., Z4 are independent, then Var (Z₁ + Z2 + Z3+ Z4) = Var (Z₁) + Var (Z₂) + Var (Z3) + Var (Z₁). These methods can be applied to obtain even more general results, such as Var (1Z₁) = Var (Z₁) or Var (Σ1CZ) = Σ₁c²Var (Z₁). i=1
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.11P
Related questions
Question
Transcribed Image Text:Problem 4. This is an exercise about properties of expectations and variances, which we will use
extensively in this course. It is closely related to SW Key Concept 2.3.
Suppose you only know the following facts of variances and covariances: If X, Y, and Z are any
random variables, and a and b are any constants, then
I
Var (X+Y)
Var (aX)
Cov (aX, bY)
Cov (X+Y,Z)
Cov(a, X)
=
=
=
=
=
Var (X) + Var (Y) + 2Cov (X, Y)
a²Var (X)
abCov (X,Y)
Cov (X, Z) + Cov (Y,Z)
0.
Also, you know that if X and Y are independent (I will denote that by X LY) then Cov (X, Y) = 0.
For calculations below, please provide precise justification for every step / equality in terms
of the given information, and avoid intuitive generalizations.
(a) Use the above facts to show another related fact: If Z₁ and Z2 are random variables, and C1 and
C2 are constants, then
Var (c₁Z₁+c₂Z₂) = cVar (Z₁)+cVar (Z₂)+c1c22Cov (Z₁, Z₂).
(b) Use any of the above facts to show that if Z₁ and Z₂ are independent,
Var (Z₁ + Z2) = Var (Z₁) + Var (Z₂).
(c) Use the result in (b) repeatedly to show that if Z₁,..., Z4 are independent, then
Var (Z₁ + Z2 + Z3+ Z4) = Var (Z₁) + Var (Z₂) + Var (Z3) + Var (Z₁).
These methods can be applied to obtain even more general results, such as Var (1Z₁) =
Var (Z₁) or Var (Σ1CZ) = Σ₁c²Var (Z₁).
i=1
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