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a) Find the nth moment of X, about the origin. b) Let Y = 3X₁ + X₂. Find the second moment about the origin of variance. c) Let Z = aX₁ + 6X2, and W = aX₁ - bX₂. Determine the condition that Z and W are uncorrelated. d) When the condition you for d in gotified on Wand 7 olgo inde

a) Find the nth moment of X, about the origin. b) Let Y = 3X₁ + X₂. Find the second moment about the origin of variance. c) Let Z = aX₁ + 6X2, and W = aX₁ - bX₂. Determine the condition that Z and W are uncorrelated. d) When the condition you for d in gotified on Wand 7 olgo inde

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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### Problem Set on Random Variables Let random variables \(X_1\) and \(X_2\) be uncorrelated and each distributed according to: \[ f_X(x) = \begin{cases} \frac{2x}{T^2} & 0 \leq x \leq T, \\ 0 & \text{otherwise} \end{cases} \] **Questions:** a) **Find the nth moment of \(X_i\) about the origin.** b) **Let \(Y = 3X_1 + X_2\). Find the second moment about the origin of \(Y\), then find its variance.** c) **Let \(Z = aX_1 + bX_2\), and \(W = aX_1 - bX_2\). Determine the condition on \(a\) and \(b\) such that \(Z\) and \(W\) are uncorrelated.** d) **When the condition you found is satisfied, are \(W\) and \(Z\) also independent? Justify fully.**
Transcribed Image Text:### Problem Set on Random Variables Let random variables \(X_1\) and \(X_2\) be uncorrelated and each distributed according to: \[ f_X(x) = \begin{cases} \frac{2x}{T^2} & 0 \leq x \leq T, \\ 0 & \text{otherwise} \end{cases} \] **Questions:** a) **Find the nth moment of \(X_i\) about the origin.** b) **Let \(Y = 3X_1 + X_2\). Find the second moment about the origin of \(Y\), then find its variance.** c) **Let \(Z = aX_1 + bX_2\), and \(W = aX_1 - bX_2\). Determine the condition on \(a\) and \(b\) such that \(Z\) and \(W\) are uncorrelated.** d) **When the condition you found is satisfied, are \(W\) and \(Z\) also independent? Justify fully.**
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