As in an earlier exercise, we assume that the random variables Θ and X are described by a joint PDF which is uniform on the triangular set defined by the constraints 0≤x≤1 , 0≤θ≤x .

Question

As in an earlier exercise, we assume that the random variables  Θ  and  X  are described by a joint PDF which is uniform on the triangular set defined by the constraints  0≤x≤1 ,  0≤θ≤x .

a) Find an expression for the conditional mean squared error of the LMS estimator given that  X=x , valid for  x∈[0,1] . Express your answer in terms of  x  using standard notation.

Anonymous · Accepted Answer

The b part is 1/24. Please take note. Thank you for the answer

Anonymous · Answer

a) the conditional PDF of Θ is uniform on the range [0,x] . Hence, the conditional variance is x2/12

b)This is given by the integral of the conditional variance, weighted by the PDF of X . The PDF of X is found using the formula for going from the joint to the marginal, and is fX(x)=2x , for x∈[0,1] . Thus, the mean squared error is

∫10x212⋅2xdx=16∫10x3dx=124.

Anonymous · Answer

I know to get from the joint to the marginal we integrate, but what is the joint?

Anonymous · Answer

Part a: (x^2)/12