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15. (a) A finite family B₁, ie I of o-algebras is independent iff for every choice of non-negative Bi-measurable random variable Y₁, i € I, we have E(ΠΥ;) = ΠΕ(;). - iel iel (One direction is immediate. For the opposite direction, prove the result first for positive simple functions and then extend.) (b) If {B,,t e T) is an arbitrary independent family of o-algebras in (S2, B, P), the family (B, te T) is again independent if B, B₁, (t e T). Deduce from this that {f(X), t e T) is a family of independent random variables if the family {X,, te T) is independent and the f, are measurable. In order for the family (X₁, t e T) of random variables to be independent, it is necessary and sufficient that

15. (a) A finite family B₁, ie I of o-algebras is independent iff for every choice of non-negative Bi-measurable random variable Y₁, i € I, we have E(ΠΥ;) = ΠΕ(;). - iel iel (One direction is immediate. For the opposite direction, prove the result first for positive simple functions and then extend.) (b) If {B,,t e T) is an arbitrary independent family of o-algebras in (S2, B, P), the family (B, te T) is again independent if B, B₁, (t e T). Deduce from this that {f(X), t e T) is a family of independent random variables if the family {X,, te T) is independent and the f, are measurable. In order for the family (X₁, t e T) of random variables to be independent, it is necessary and sufficient that

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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15. (a) A finite family B₁, i EI of o-algebras is independent iff for every choice of non-negative Bi-measurable random variable Y₁, i E I, we have E(Πx) = ΠΕ(;). iel iel (One direction is immediate. For the opposite direction, prove the result first for positive simple functions and then extend.) (b) If (B₁, t e T) is an arbitrary independent family of o-algebras in (S2, B, P), the family (B,, te T) is again independent if B, B, (t = T). Deduc this that {f(X), te T) is a family of independent random variables if the family (X,, te T) is independent and the f, are measurable. In order for the family {X,, te T) of random variables to be independent, it is necessary and sufficient that E (П») - for every finite family {fj, je J} of bounded measurable functions. E (ƒj (X;))
Transcribed Image Text:15. (a) A finite family B₁, i EI of o-algebras is independent iff for every choice of non-negative Bi-measurable random variable Y₁, i E I, we have E(Πx) = ΠΕ(;). iel iel (One direction is immediate. For the opposite direction, prove the result first for positive simple functions and then extend.) (b) If (B₁, t e T) is an arbitrary independent family of o-algebras in (S2, B, P), the family (B,, te T) is again independent if B, B, (t = T). Deduc this that {f(X), te T) is a family of independent random variables if the family (X,, te T) is independent and the f, are measurable. In order for the family {X,, te T) of random variables to be independent, it is necessary and sufficient that E (П») - for every finite family {fj, je J} of bounded measurable functions. E (ƒj (X;))
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