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.
The b part is 1/24. Please take note. Thank you for the answer
Part a: (x^2)/12
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.
I know to get from the joint to the marginal we integrate, but what is the joint?
Why did the chicken go to the therapist's office? Because it had a conditional mean squared error and needed some standard notation to express itself!
To find the expression for the conditional mean squared error (CMSE) of the LMS (Least Mean Squares) estimator given that X=x, we need to compute the variance of the estimate error. The LMS estimator can be denoted as:
Ŷ = E[Θ|X=x]
Where Ŷ is the estimated value of Θ given X=x.
To calculate the CMSE, we need to find the variance of the estimation error, which is given by:
Var(Θ - Ŷ)
To find this variance, we'll break it down into its components. The estimation error can be written as:
Θ - Ŷ = Θ - E[Θ|X=x]
Now, let's calculate the conditional mean squared error of the LMS estimator:
CMSE(x) = E[(Θ - Ŷ)^2|X=x]
Using the law of total expectation, we can rewrite CMSE(x) as:
CMSE(x) = E[E[(Θ - Ŷ)^2|X=x]]
Since we are assuming that the joint PDF is uniform on the triangular set defined by the constraints 0≤x≤1 and 0≤θ≤x, we know that the joint PDF can be represented as:
f(θ, x) = 2
Given this joint PDF, we can find the conditional PDF of Θ given X=x by normalizing the joint PDF over the range 0≤θ≤x. The conditional PDF is:
f(θ|X=x) = f(θ, x) / ∫[0,x] f(θ, x) dθ = 2 / (2x^2) = 1 / x^2
Now, we can calculate the conditional mean squared error by integrating the squared estimation error multiplied by the conditional PDF:
CMSE(x) = E[E[(Θ - Ŷ)^2|X=x]]
= E[E[(Θ - E[Θ|X=x])^2|X=x]]
= E[E[(Θ - x/2)^2|X=x]]
= ∫[0,x] (θ - x/2)^2 * (1 / x^2) dθ
Simplifying the integrand:
CMSE(x) = ∫[0,x] (θ^2 - 2θ(x/2) + (x/2)^2) / x^2 dθ
= ∫[0,x] (θ^2 - θx + x^2/4) / x^2 dθ
= ∫[0,x] θ^2/x^2 - θ/x + (x^2/4) / x^2 dθ
= (1/x^2) ∫[0,x] θ^2 dθ - (1/x) ∫[0,x] θ dθ + (1/x^2) ∫[0,x] (x^2/4) dθ
= (1/x^2) [(θ^3/3) |[0,x] - (θ^2/2) |[0,x] + (x^2/4)θ |[0,x]]
= (1/x^2) [(x^3/3) - (x^2/2) + (x^3/4)]
= (1/x^2) [(4x^3/12) - (6x^2/12) + (3x^3/12)]
= (1/x^2) [(x^3/12)]
= (1/12) * x
Therefore, the expression for the conditional mean squared error of the LMS estimator, given that X=x, is (1/12) * x.