# Homework Help: Conditional Expectation Problem

1. Mar 6, 2012

### ixtabai

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...