5.21 Given a probability space (S,F,P) and a random variable X. Let A be a sub-o-field of F, and consider Â = E (X | A). (i) Show that E ((X – Â)Y) = 0 if Y is an A-measurable random vari- able. (ii) Let Y run over all A-measurable random variables. Show that the squared error E ((X – Y)²) is minimal when Y = X.

5.21 Given a probability space (S,F,P) and a random variable X. Let A be a sub-o-field of F, and consider Â = E (X | A). (i) Show that E ((X – Â)Y) = 0 if Y is an A-measurable random vari- able. (ii) Let Y run over all A-measurable random variables. Show that the squared error E ((X – Y)²) is minimal when Y = X.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.CR: Chapter 13 Review

Problem 22CR

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