Exercise 13.4 Let X₁, **** (a) Show that X, be independent random variables with expected values E[X] =μ, and consider the following simulation estimator of E[Y]: n W=Y+(X₁₁). i=1 - n n Var(W) = Var(Y) + c²/ Var(X;) +2Σ c; Cov(Y, X;). i=1 i=1 (b) Use calculus to show that the values of c₁, ..., c, that minimize Var(W) are Cov(Y, X;) Ci i=1,..., R. Var (X)
Exercise 13.4 Let X₁, **** (a) Show that X, be independent random variables with expected values E[X] =μ, and consider the following simulation estimator of E[Y]: n W=Y+(X₁₁). i=1 - n n Var(W) = Var(Y) + c²/ Var(X;) +2Σ c; Cov(Y, X;). i=1 i=1 (b) Use calculus to show that the values of c₁, ..., c, that minimize Var(W) are Cov(Y, X;) Ci i=1,..., R. Var (X)
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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![Exercise 13.4 Let X₁,
****
(a) Show that
X, be independent random variables with expected values E[X] =μ, and consider the following simulation estimator of E[Y]:
n
W=Y+(X₁₁).
i=1
-
n
n
Var(W) = Var(Y) + c²/ Var(X;) +2Σ c; Cov(Y, X;).
i=1
i=1
(b) Use calculus to show that the values of c₁, ..., c, that minimize Var(W) are
Cov(Y, X;)
Ci
i=1,..., R.
Var (X)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b346ae-ef24-4d99-babf-e95763a360cd%2F52cbeadc-8f44-4943-9bb5-0ea0117f80ca%2Fpkcitb7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 13.4 Let X₁,
****
(a) Show that
X, be independent random variables with expected values E[X] =μ, and consider the following simulation estimator of E[Y]:
n
W=Y+(X₁₁).
i=1
-
n
n
Var(W) = Var(Y) + c²/ Var(X;) +2Σ c; Cov(Y, X;).
i=1
i=1
(b) Use calculus to show that the values of c₁, ..., c, that minimize Var(W) are
Cov(Y, X;)
Ci
i=1,..., R.
Var (X)
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