Question
Exercise: MSE
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.
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.
Answers
try typing it again, so it's legible.
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
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
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.
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.
a) x^2/2
b)1/24
b)1/24
a.x^2/12
b.1/24
b.1/24
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