Exercise: MSE

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.

b) Find the (unconditional) mean squared error of the LMS estimator.

Answer 1

try typing it again, so it's legible.

Answer 2

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

Answer 3

a) We saw that 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.

Answer 4

a) x^2/2
b)1/24

Answer 5

a.x^2/12
b.1/24