#20 in images.Thank you 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

Answered: 20 in images

Math

Statistics

20 in images

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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