Suppose that the random variables Θ and X are not independent, but E[Θ∣X=x]=3 for all x . Then the LLMS estimator of Θ based on X is of the form aX+b , with
a=
b=
a=0
b=3
such that E[Θ]=ax+b
E[Θ]=3 for all x therefore 0(x)+3=3 for all x
a = "a clown trying to balance on a unicycle while juggling flaming torches."
b = "a rubber chicken that squawks every time you squeeze it."
To find the LLMS (Linear Least Mean Squares) estimator of Θ based on X, let's start by writing the equation in the form Θ = aX + b, where a and b are constants to be determined.
Given that E[Θ∣X=x] = 3 for all x, we can rewrite this as:
E[Θ - 3X∣X=x] = 0 [subtract 3X from both sides]
Now, since E[Θ - 3X∣X=x] = 0, this implies that the covariance between Θ - 3X and X is zero. So we have:
Cov(Θ - 3X, X) = 0
Expanding the covariance, we get:
Cov(Θ, X) - 3Cov(X, X) = 0
Since Cov(X, X) is the variance of X (Var(X)), and Cov(Θ, X) is the covariance between Θ and X (Cov(Θ, X)), we can rewrite the above equation as:
Cov(Θ, X) - 3Var(X) = 0
Now, let's assume that X and Θ have finite variances (Var(X) ≠ ∞ and Var(Θ) ≠ ∞). In that case, we can solve for Cov(Θ, X):
Cov(Θ, X) = 3Var(X)
From this equation, we can see that Cov(Θ, X) is equal to 3 times the variance of X.
Finally, we know that the LLMS estimator of Θ based on X is given by:
Î(Θ) = Cov(Θ, X) / Var(X)
Substituting the values, we have:
Î(Θ) = 3Var(X) / Var(X) = 3
Therefore, the LLMS estimator of Θ based on X is simply 3.
To find the values of a and b in the LLMS estimator of Θ based on X, we can use the property of conditional expectation and the concept of the best linear unbiased estimator (BLUE).
Since E[Θ∣X=x] = 3 for all x, this implies that the conditional expectation of Θ given X is a constant value of 3. Therefore, the best linear unbiased estimator of Θ based on X should also be a constant.
To find this constant value, we can use the property of conditional expectation:
E[Θ] = E[E[Θ∣X]].
Since E[Θ∣X] is a constant, we can take it out of the inner expectation:
E[Θ] = E[3] = 3.
Therefore, the LLMS estimator of Θ based on X is a constant equal to 3.
In this case, a = 0 and b = 3.