3. Let X and Y have the joint distribution as indicated in the following table: X=0 X=1 |Y=0 1/3 0 Y=1 0 1/3 Y=2 1/3 0 a. Give Y, the linear MMSE estimate for Y given X. b. Give the MSE for Ŷ₁. c. Give YM, the MMSE estimate for Y given X. d. Give the MSE for ÎM.
3. Let X and Y have the joint distribution as indicated in the following table: X=0 X=1 |Y=0 1/3 0 Y=1 0 1/3 Y=2 1/3 0 a. Give Y, the linear MMSE estimate for Y given X. b. Give the MSE for Ŷ₁. c. Give YM, the MMSE estimate for Y given X. d. Give the MSE for ÎM.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The image contains a problem and a table related to joint distribution in statistics. Here’s the transcription:
---
3. Let \( X \) and \( Y \) have the joint distribution as indicated in the following table:
\[
\begin{array}{c|ccc}
& Y=0 & Y=1 & Y=2 \\
\hline
X=0 & 1/3 & 0 & 1/3 \\
X=1 & 0 & 1/3 & 0 \\
\end{array}
\]
a. Give \(\hat{Y}_L\), the linear MMSE (Minimum Mean Square Error) estimate for \( Y \) given \( X \).
b. Give the MSE (Mean Square Error) for \(\hat{Y}_L\).
c. Give \(\hat{Y}_M\), the MMSE estimate for \( Y \) given \( X \).
d. Give the MSE for \(\hat{Y}_M\).
---
**Explanation of the Table:**
The table presents the joint probability distribution of two random variables, \( X \) and \( Y \). The rows of the table represent different values of \( X \), while the columns represent different values of \( Y \). Each cell in the table shows the probability that \( X \) and \( Y \) take on the values specified by the respective row and column.
- For \( X = 0 \), \( Y = 0 \) has a probability of \( \frac{1}{3} \), \( Y = 1 \) has a probability of \( 0 \), and \( Y = 2 \) has a probability of \( \frac{1}{3} \).
- For \( X = 1 \), \( Y = 0 \) has a probability of \( 0 \), \( Y = 1 \) has a probability of \( \frac{1}{3} \), and \( Y = 2 \) has a probability of \( 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa86feb48-aed4-4133-a748-653f8a12a813%2F9de852ba-b991-44a8-804c-2f47a989bc18%2Fjyymbne_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a problem and a table related to joint distribution in statistics. Here’s the transcription:
---
3. Let \( X \) and \( Y \) have the joint distribution as indicated in the following table:
\[
\begin{array}{c|ccc}
& Y=0 & Y=1 & Y=2 \\
\hline
X=0 & 1/3 & 0 & 1/3 \\
X=1 & 0 & 1/3 & 0 \\
\end{array}
\]
a. Give \(\hat{Y}_L\), the linear MMSE (Minimum Mean Square Error) estimate for \( Y \) given \( X \).
b. Give the MSE (Mean Square Error) for \(\hat{Y}_L\).
c. Give \(\hat{Y}_M\), the MMSE estimate for \( Y \) given \( X \).
d. Give the MSE for \(\hat{Y}_M\).
---
**Explanation of the Table:**
The table presents the joint probability distribution of two random variables, \( X \) and \( Y \). The rows of the table represent different values of \( X \), while the columns represent different values of \( Y \). Each cell in the table shows the probability that \( X \) and \( Y \) take on the values specified by the respective row and column.
- For \( X = 0 \), \( Y = 0 \) has a probability of \( \frac{1}{3} \), \( Y = 1 \) has a probability of \( 0 \), and \( Y = 2 \) has a probability of \( \frac{1}{3} \).
- For \( X = 1 \), \( Y = 0 \) has a probability of \( 0 \), \( Y = 1 \) has a probability of \( \frac{1}{3} \), and \( Y = 2 \) has a probability of \( 0 \).
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