What is the LMS Estimate of Theta for a Joint PDF on a Triangular Set?

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In summary, the conversation discusses finding the LMS estimate of theta given a joint PDF that is uniform on a triangular set with certain constraints. The process involves calculating the joint PDF and integrating to find the expected value of theta, which is the midpoint between 0 and x.
  • #1
Sitingbull
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I have the following question and I am struggling to find the right answer.

The random variables and 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 in the range [0,1] . Express your answer in terms x.

I started by calculating the joint pdf by first calculating the area of the triangle which is 1/2 * x * 1 = x /2 . The joint pdf will be 1 / (x/2) = 2 / x

Then I integrate over integral over (x to 1) of theta times 2 / x. I got an integral of \theta^ 2 / x which gives me a final answer of (1-X^2) / x

Does that look good or I am missing something ...

Sorry for my notation which lacks LATEX, I am new here.Thank youSB
 
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  • #2
Sitingbull said:
I have the following question and I am struggling to find the right answer.

The random variables and 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 in the range [0,1] . Express your answer in terms x.

I started by calculating the joint pdf by first calculating the area of the triangle which is 1/2 * x * 1 = x /2 .
What triangle are you talking about? I assumed you meant the triangle given here, 0<= x<= 1, 0<= theta<= x but the area of that does not depend on x! The area is 1/2.

The joint pdf will be 1 / (x/2) = 2 / x
Further, The probability density function is exactly what is said above- a constant since this is a "uniform distribution": 2 for all x, theta in the regon.

Then I integrate over integral over (x to 1) of theta times 2 / x. I got an integral of \theta^ 2 / x which gives me a final answer of (1-X^2) / x

Does that look good or I am missing something ...

Sorry for my notation which lacks LATEX, I am new here.Thank youSB
Given that x= X, a fixed value, theta can rang from 0 up to X. Take the square root of the integral of Theta^2 from 0 to X.
 
  • #3
Thank you Country Boy you are totally right, I managed at the end to get it. Its simply the expecation of theta which is the midpoint between 0 and x. Thank you a lot
 

1. What is the Least Mean Square (LMS) algorithm?

The Least Mean Square (LMS) algorithm is a type of adaptive filtering algorithm used to minimize the mean square error between the desired output and the actual output of a system. It is commonly used in signal processing and machine learning applications.

2. How does the LMS algorithm work?

The LMS algorithm works by iteratively adjusting the weights of a linear filter in order to minimize the mean square error. This is done by using an error signal calculated from the difference between the desired output and the actual output of the system. The algorithm then uses this error signal to update the filter weights in a way that reduces the error in future iterations.

3. What are the advantages of using the LMS algorithm?

The LMS algorithm is a simple and efficient way to adapt to changing environments or input signals. It is also computationally efficient, making it suitable for real-time applications. Additionally, the LMS algorithm does not require knowledge of the underlying system model, making it a useful tool for unknown or complex systems.

4. What are the limitations of the LMS algorithm?

One of the main limitations of the LMS algorithm is its sensitivity to noisy data. If the input signals are highly correlated or contain a lot of noise, the algorithm may not converge to the optimal solution. Additionally, the LMS algorithm may converge slowly in certain situations, requiring a large number of iterations to reach an acceptable error level.

5. How is the LMS algorithm used in practical applications?

The LMS algorithm has a wide range of applications, including adaptive noise cancellation, channel equalization, and adaptive beamforming. It is also commonly used in machine learning for tasks such as prediction and classification. In these applications, the LMS algorithm is used to update the parameters of a model in order to improve its performance over time.

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