Probability

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.

  1. 👍 0
  2. 👎 3
  3. 👁 1,137
  1. try typing it again, so it's legible.

    1. 👍 0
    2. 👎 5
  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

    1. 👍 0
    2. 👎 0
  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.

    1. 👍 0
    2. 👎 0
  4. a) x^2/2
    b)1/24

    1. 👍 0
    2. 👎 0
  5. a.x^2/12
    b.1/24

    1. 👍 2
    2. 👎 0

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