1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Show that the expectation is zero

  1. Sep 12, 2006 #1
    In one of the proofs I'm reading, I need to show that the
    that will minimize [tex]E[Y - \phi(X)]^{2}[/tex] is
    [tex]\phi(X) = E(Y|X)[/tex]

    the proof is shown to proceed as follows:
    [tex]E[Y - \phi(X)]^{2}[/tex]
    [tex]=E[(Y - E(Y|X)) + (E(Y|X) - \phi(X))]^{2}[/tex]

    expanding, we have

    [tex]=E[(Y - E(Y|X))^{2} + 2(Y - E(Y|X))(E(Y|X) - \phi(X)) + (E(Y|X) - \phi(X))^{2}][/tex]
    [tex]=E[(Y - E(Y|X))^{2}] + 2E[(Y - E(Y|X))(E(Y|X) - \phi(X))] + E[(E(Y|X) - \phi(X))]^{2}][/tex]

    the proof then states that the middle term is equal to zero. How is this true?
    I got
    [tex]2E[(Y - E(Y|X))(E(Y|X) - \phi(X))] [/tex]
    [tex]=2E[YE(Y|X) - (E(Y|X))^{2} - Y\phi(X) + E(Y|X)\phi(X)][/tex]
    [tex]=2E[YE(Y|X)] - 2E[(E(Y|X))^{2}] - 2E[Y\phi(X)] + 2E[E(Y|X)\phi(X)][/tex]
    [tex]=2E(Y|X)E(Y) - 2(E(Y|X))^{2} - 2E[Y\phi(X)] + 2E(Y|X)E(\phi(X))[/tex]

    I'm not sure on how to proceed next.
    I'm thinking that the first 2 terms will cancel as will the last two...but how exactly I'm not sure.

    help please. thanks.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Threads for Show expectation zero Date
Showing that one polynomial divides another Wednesday at 11:04 PM
Show that a series is divergent Apr 7, 2018
Showing that an inequality is true Mar 29, 2018
Need help showing this difference quotient is a derivative Mar 20, 2018
Expected Monthly Profit Jun 16, 2017