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Conditional Expectation Problem

  1. Mar 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose that X and Y have a continuous joint distribution with joint pdf given by
    f (x, y) = { x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
    0 otherwise.

    Suppose that a person can pay a cost c for the opportunity of observing the value of
    X before predicting the value of Y or can simply predict the value of Y without first observing the value of X . If the total loss is considered to be the cost c plus the mean square error (MSE) of the predicted value, what is the maximum value of c that one
    should be willing to pay?

    2. Relevant equations

    Conditional Expectation: E[Y given X=x] and E[C]

    3. The attempt at a solution

    Broke the problem into 2 parts: Part 1 is the Conditional Expectation of predicting Y by first observing X; and Part 2 simply predicting Y
    1. Found the Conditional Expectation of Y given X to be 2/3 and the MMSE (Variance) 1/18
    2. Found the Expectation of Y to be 7/12 and MMSE (Variance) 11/144

    Given this is even the way to go I'm stuck as to where to go next.

    In the problem is says Total Loss = c + MMSE Could this be TL= c + 19/144? This can't be this simple...

    Thanks for your help.
     
    Last edited: Mar 6, 2012
  2. jcsd
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