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)
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.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a579a7f-99c6-465f-8352-06e32f8837ce%2F07b0a500-0d8e-45e1-a3f6-069caacdb76a%2F6715b6t_processed.png&w=3840&q=75)
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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