Let X and Y be jointly continuous nonnegative random variables. A particular value y of Y is observed and it turns out that fX|Y(x∣y)=2e−2x , for x≥0 .

1. Find the LMS estimate (conditional expectation) of X .


2. Find the conditional mean squared error E[(X−XˆLMS)2∣Y=y] .

3. Find the MAP estimate of X .

4. Find the conditional mean squared error E[(X−XˆMAP)2∣Y=y] .