• Given two random variables X and Y, both with nonzero variance, define the function h(t) = E[((X — ux)+t(Y – My))]. (a) Show that h(t) ≥ 0. (b) Show that h(t) = Var(X) + 2 Cov(X, Y)t + Var(Y)t² (c) h is a nonnegative parabola, so it has at most one root. Thus its discriminant is at most 0. Use this to deduce that the correlation coefficient p satisfies -1 ≤p ≤ 1.
• Given two random variables X and Y, both with nonzero variance, define the function h(t) = E[((X — ux)+t(Y – My))]. (a) Show that h(t) ≥ 0. (b) Show that h(t) = Var(X) + 2 Cov(X, Y)t + Var(Y)t² (c) h is a nonnegative parabola, so it has at most one root. Thus its discriminant is at most 0. Use this to deduce that the correlation coefficient p satisfies -1 ≤p ≤ 1.
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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Random variables and variance
![- Given two random variables \( X \) and \( Y \), both with nonzero variance, define the function \( h(t) = \mathbb{E}[(X - \mu_X) + t(Y - \mu_Y))^2] \).
(a) Show that \( h(t) \geq 0 \).
(b) Show that \( h(t) = \text{Var}(X) + 2 \text{Cov}(X, Y) t + \text{Var}(Y) t^2 \).
(c) \( h \) is a nonnegative parabola, so it has at most one root. Thus its discriminant is at most 0. Use this to deduce that the correlation coefficient \( \rho \) satisfies \(-1 \leq \rho \leq 1\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F1034038a-de23-46e6-ad47-931a7a172a7a%2Fwys2y6q_processed.png&w=3840&q=75)
Transcribed Image Text:- Given two random variables \( X \) and \( Y \), both with nonzero variance, define the function \( h(t) = \mathbb{E}[(X - \mu_X) + t(Y - \mu_Y))^2] \).
(a) Show that \( h(t) \geq 0 \).
(b) Show that \( h(t) = \text{Var}(X) + 2 \text{Cov}(X, Y) t + \text{Var}(Y) t^2 \).
(c) \( h \) is a nonnegative parabola, so it has at most one root. Thus its discriminant is at most 0. Use this to deduce that the correlation coefficient \( \rho \) satisfies \(-1 \leq \rho \leq 1\).
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