lculate Var(X). When is the variance finite? nd the n-th moment of X. What is the maximum n such that the n-th moment is ite? ppose X follows the power law with p = 2 (has PDF f(x) = x2 for x ≥ 1) and opose Y = X-2. Find the PDF of Y.
lculate Var(X). When is the variance finite? nd the n-th moment of X. What is the maximum n such that the n-th moment is ite? ppose X follows the power law with p = 2 (has PDF f(x) = x2 for x ≥ 1) and opose Y = X-2. Find the PDF of Y.
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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![### Calculate Var(X). When is the variance finite?
To calculate the variance of \( X \), denoted Var(\( X \)), we need to determine the expected values \( E(X) \) and \( E(X^2) \). The variance is given by:
\[ \text{Var}(X) = E(X^2) - (E(X))^2 \]
We then need to analyze under what conditions these expected values exist and are finite.
### Find the \( n \)-th moment of \( X \). What is the maximum \( n \) such that the \( n \)-th moment is finite?
The \( n \)-th moment of a random variable \( X \) is defined as:
\[ E(X^n) \]
We will calculate this moment and determine the largest integer \( n \) for which this moment is finite, denoted by determining when \( E(X^n) \) converges.
### Suppose \( X \) follows the power law with \( \rho = 2 \)
If \( X \) follows a power law distribution with \( \rho = 2 \), then the probability density function (PDF) of \( X \) is given by:
\[ f(x) = x^{-2} \quad \text{for} \quad x \geq 1 \]
### Suppose \( Y = X^{-2} \). Find the PDF of \( Y \).
When transforming the random variable \( X \) to \( Y = X^{-2} \), we aim to find the PDF of \( Y \). This involves using the change of variables technique and the relationship between the densities of \( X \) and \( Y \).
By following these steps, we will provide detailed solutions and explanations to understand the underlying principles and calculations required for these problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F153bfb9c-b05e-4b91-94e3-9e151aaf7f28%2Fa0a1c50d-df40-4708-9233-8a0ab0d7cda3%2Fb6mrwrs_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculate Var(X). When is the variance finite?
To calculate the variance of \( X \), denoted Var(\( X \)), we need to determine the expected values \( E(X) \) and \( E(X^2) \). The variance is given by:
\[ \text{Var}(X) = E(X^2) - (E(X))^2 \]
We then need to analyze under what conditions these expected values exist and are finite.
### Find the \( n \)-th moment of \( X \). What is the maximum \( n \) such that the \( n \)-th moment is finite?
The \( n \)-th moment of a random variable \( X \) is defined as:
\[ E(X^n) \]
We will calculate this moment and determine the largest integer \( n \) for which this moment is finite, denoted by determining when \( E(X^n) \) converges.
### Suppose \( X \) follows the power law with \( \rho = 2 \)
If \( X \) follows a power law distribution with \( \rho = 2 \), then the probability density function (PDF) of \( X \) is given by:
\[ f(x) = x^{-2} \quad \text{for} \quad x \geq 1 \]
### Suppose \( Y = X^{-2} \). Find the PDF of \( Y \).
When transforming the random variable \( X \) to \( Y = X^{-2} \), we aim to find the PDF of \( Y \). This involves using the change of variables technique and the relationship between the densities of \( X \) and \( Y \).
By following these steps, we will provide detailed solutions and explanations to understand the underlying principles and calculations required for these problems.
![### Power Laws
A random variable \( X \) follows the power law with parameter \( \rho > 0 \) when \( X \) has a probability density function (PDF) given by
\[ f_X(x) = c_{\rho} x^{-\rho} \ \text{for} \ x \geq 1 \]
where \( \rho \) governs how fast the probabilities go to 0 as \( x \) goes to \( \infty \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F153bfb9c-b05e-4b91-94e3-9e151aaf7f28%2Fa0a1c50d-df40-4708-9233-8a0ab0d7cda3%2Fsxnxzfeb_processed.png&w=3840&q=75)
Transcribed Image Text:### Power Laws
A random variable \( X \) follows the power law with parameter \( \rho > 0 \) when \( X \) has a probability density function (PDF) given by
\[ f_X(x) = c_{\rho} x^{-\rho} \ \text{for} \ x \geq 1 \]
where \( \rho \) governs how fast the probabilities go to 0 as \( x \) goes to \( \infty \).
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