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MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![The text appears to be from a textbook on probability or statistics, dealing with conditional expectation and correlation coefficients. Below is the transcription suitable for an educational website:
---
### Conditional Expectation
**Find the correlation coefficient between \( Z \) and \( W \) and show that**
\[
0 \leq \rho^2 \leq \left( \frac{\sigma_1^2 - \sigma_2^2}{\sigma_1^2 + \sigma_2^2} \right)^2,
\]
where \( \rho \) denotes the correlation coefficient between \( Z \) and \( W \).
**19.** Let \( (X_1, X_2, \ldots, X_n) \) be an RV such that the correlation coefficient between each pair \( X_i, X_j, i \neq j, \) is \( \rho \). Show that \(- (n - 1)^{-1} \leq \rho \leq 1\).
**20.** Let \( X_1, X_2, \ldots, X_{m+n} \) be iid RVs with finite second moment. Let
\[
S_k = \sum_{j=1}^{k} X_j, \, k = 1, 2, \ldots, m+n.
\]
Find the correlation coefficient between \( S_n \) and \( S_{m+n} - S_m \), where \( n > m \).
*Note:* There is a handwritten note showing
\[
\rho = \frac{m - n}{n}.
\]
**21.** Let \( f \) be the PDF of a positive RV, and write
\[
g(x, y) =
\begin{cases}
\frac{f(x+y)}{x+y} & \text{if } x > 0, y > 0, \\
0 & \text{otherwise}.
\end{cases}
\]
Show that \( g \) is a density function in the plane. If the \( m \)th moment of \( f \) exists for some positive integer \( m \), find \( EX^m \). Compute the means and variances of \( X \) and \( Y \) and the correlation coefficient between \( X \) and \( Y \) in terms of moments of \( f \). (Adapted from](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f1ebd08-7e04-46cf-88d9-0b90774940c7%2F50101750-69ca-461e-836f-81ab7837f5cd%2Fb6njjmd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The text appears to be from a textbook on probability or statistics, dealing with conditional expectation and correlation coefficients. Below is the transcription suitable for an educational website:
---
### Conditional Expectation
**Find the correlation coefficient between \( Z \) and \( W \) and show that**
\[
0 \leq \rho^2 \leq \left( \frac{\sigma_1^2 - \sigma_2^2}{\sigma_1^2 + \sigma_2^2} \right)^2,
\]
where \( \rho \) denotes the correlation coefficient between \( Z \) and \( W \).
**19.** Let \( (X_1, X_2, \ldots, X_n) \) be an RV such that the correlation coefficient between each pair \( X_i, X_j, i \neq j, \) is \( \rho \). Show that \(- (n - 1)^{-1} \leq \rho \leq 1\).
**20.** Let \( X_1, X_2, \ldots, X_{m+n} \) be iid RVs with finite second moment. Let
\[
S_k = \sum_{j=1}^{k} X_j, \, k = 1, 2, \ldots, m+n.
\]
Find the correlation coefficient between \( S_n \) and \( S_{m+n} - S_m \), where \( n > m \).
*Note:* There is a handwritten note showing
\[
\rho = \frac{m - n}{n}.
\]
**21.** Let \( f \) be the PDF of a positive RV, and write
\[
g(x, y) =
\begin{cases}
\frac{f(x+y)}{x+y} & \text{if } x > 0, y > 0, \\
0 & \text{otherwise}.
\end{cases}
\]
Show that \( g \) is a density function in the plane. If the \( m \)th moment of \( f \) exists for some positive integer \( m \), find \( EX^m \). Compute the means and variances of \( X \) and \( Y \) and the correlation coefficient between \( X \) and \( Y \) in terms of moments of \( f \). (Adapted from
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