Conditional Expectation Problem

[ixtabai]

Homework Statement

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?

Homework Equations

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

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.

 

[mighty2000]

Hello,

Thank you for your post. It seems like you have made some progress in solving this problem. However, there are a few things that can be clarified and improved upon in your solution.

Firstly, when calculating the conditional expectation of Y given X, you should use the formula E[Y|X] = ∫y*f(y|x)dy, where f(y|x) is the conditional pdf of Y given X. In this case, the conditional pdf is simply (x+y), as given in the problem. Therefore, the conditional expectation becomes E[Y|X] = ∫y*(x+y)dy from 0 to 1, which evaluates to (x+2)/3. This is the correct answer for Part 1 of your solution.

For Part 2, you have correctly found the expectation of Y, but the MMSE (mean squared error) should be calculated using the formula MMSE = E[(Y - E[Y])^2]. This evaluates to 11/144, which is the correct answer for Part 2.

Now, for the total loss, you are correct in thinking that it is TL = c + MMSE. However, you should use the values you have calculated for Part 1 and Part 2, which are (x+2)/3 and 11/144, respectively. Therefore, TL = c + (x+2)/3 + 11/144.

To find the maximum value of c that one should be willing to pay, you can set the derivative of TL with respect to c equal to 0 and solve for c. This will give you the optimal value of c that minimizes the total loss. I hope this helps! Let me know if you have any further questions.