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.
To find the conditional mean squared error (MSE) of the LMS estimator given that X=x, we need to use the definition of MSE and integrate over the joint PDF of Θ and X. The conditional mean squared error can be calculated as follows:
a) Conditional Mean Squared Error:
MSE = E[(θ - θ_hat)^2 | X=x]
Since we are given that the joint PDF is uniform on the triangular set defined by the constraints 0≤x≤1, 0≤θ≤x, we can rewrite the joint PDF as:
f(θ, x) = 2
Now, let's calculate the conditional mean squared error using the formula for MSE:
MSE = ∫∫(θ - θ_hat)^2 * f(θ, x) dθ dx
Since the range of θ is from 0 to x, we can rewrite the integral as:
MSE = ∫[0,x]∫[0,x](θ - θ_hat)^2 * 2 dθ dx
To further simplify this equation, we need to calculate θ_hat, which is the LMS estimator. Without the specific form of the estimator, it's not possible to provide an expression for the conditional MSE in terms of x using standard notation.
b) Unconditional Mean Squared Error:
To find the unconditional mean squared error of the LMS estimator, we need to average the conditional MSE over all possible values of X. Since X is also uniformly distributed on the interval [0, 1], we can calculate the unconditional MSE as follows:
MSE_unconditional = E[(θ - θ_hat)^2]
MSE_unconditional = ∫[0,1]MSE * f(x) dx
where f(x) is the PDF of X, which is 1 in the interval [0, 1].
MSE_unconditional = ∫[0,1]∫[0,x](θ - θ_hat)^2 * 2 dx dθ
Again, without the specific form of the LMS estimator, it's not possible to provide a closed-form expression for the unconditional MSE using standard notation.