The random variables Theta and X are described by a joint PDF which is uniform on the triangular set defined by the constraints 0<=x<=1, 0<=theta<=x. Find the LMS estimate of Theta given that X=x, for x in the range . Express your answer in terms of x using standard notation.
To find the LMS (Least Mean Square) estimate of Theta given that X=x, we can use the conditional expectation.
The conditional expectation of Theta given that X=x, denoted as E[Theta|X=x], can be calculated using the joint PDF of Theta and X.
First, let's write the joint PDF of Theta and X:
f(theta, x) = 2, for 0<=x<=1 and 0<=theta<=x
f(theta, x) = 0, otherwise
To find the conditional expectation, we need to calculate the conditional PDF of Theta given X=x, denoted as f(theta|X=x).
The conditional PDF can be found by taking the joint PDF and dividing it by the marginal PDF of X, which is obtained by integrating the joint PDF with respect to Theta.
To find the marginal PDF of X, we integrate the joint PDF over the range of Theta:
f_X(x) = ∫[0,x] f(theta, x) d(theta)
= ∫[0,x] 2 d(theta)
= 2x
Now, we can calculate the conditional PDF f(theta|X=x):
f(theta|X=x) = f(theta, x) / f_X(x)
= (2 / (2x)) = 1 / x, for 0<=theta<=x
The conditional PDF f(theta|X=x) follows a uniform distribution from 0 to x.
To find the LMS estimate of Theta given that X=x, we take the expected value of the conditional PDF f(theta|X=x):
E[Theta|X=x] = ∫[0,x] theta * f(theta|X=x) d(theta)
= ∫[0,x] theta * (1 / x) d(theta)
= (1 / x) * ∫[0,x] theta d(theta)
= (1 / x) * (theta^2 / 2) evaluated from 0 to x
= (1 / x) * (x^2 / 2)
= x / 2
Therefore, the LMS estimate of Theta given that X=x is x/2, where x is in the range 0 to 1.
To find the LMS (Least Mean Squares) estimate of Theta given that X=x, we need to find the conditional PDF of Theta given X=x and then find the mean of this conditional PDF.
Let's start by finding the conditional PDF of Theta given X=x.
Since the joint PDF is uniform on the triangular set defined by 0<=x<=1 and 0<=theta<=x, the joint PDF can be written as:
f(x, theta) = k, for 0<=x<=1 and 0<=theta<=x
0, otherwise
To find the value of k, we can integrate the joint PDF over the entire triangular set and set it equal to 1 (since the PDF must integrate to 1):
1 = ∫∫f(x, theta) dtheta dx
= ∫∫k dtheta dx
= k * ∫(0 to x)∫(0 to 1) dtheta dx
= k * ∫(0 to x) 1 dtheta dx
= k * [theta] (0 to x)
= k * (x - 0)
= kx
Therefore, k = 1/x for 0<=x<=1 and 0<=theta<=x.
Now, let's find the conditional PDF of Theta given X=x, denoted as f(theta|x).
By definition, the conditional PDF is obtained by dividing the joint PDF by the marginal PDF of X.
The marginal PDF of X, denoted as f(x), is obtained by integrating the joint PDF over the range of theta:
f(x) = ∫f(x, theta) d(theta)
= ∫k d(theta)
= k * [theta] (0 to x)
= kx
Therefore, the conditional PDF of Theta given X=x is:
f(theta|x) = f(x, theta) / f(x)
= (1/x) / (kx)
= 1
Since the conditional PDF of Theta given X=x is uniform over the range 0<=theta<=x, the mean (LMS estimate) of this conditional PDF is simply the average of this range.
Therefore, the LMS estimate of Theta given that X=x, for x in the range, is:
E[Theta|x] = (x + 0) / 2
= x/2
Expressed in terms of x using standard notation, the LMS estimate is:
E[Theta|X=x] = x/2