### Problem StatementTo understand the proper formulation and properties of probability distributions, consider the following tasks:1. **Calculate \( c_{\rho} \) so that \( f_{X}(x) \) is a valid Probability Density Function (PDF).** - A valid PDF must satisfy the properties of being non-negative and integrating to 1 over its entire range.2. **Find the Cumulative Distribution Function (CDF) of \( X \) parameterized by \( \rho \).** - The CDF, \( F_{X}(x) \), represents the probability that the random variable \( X \) is less than or equal to a certain value \( x \) and is derived by integrating the PDF over the relevant range.3. **Find \( P(2 < X \leq 4) \).** - This probability can be found by evaluating the CDF at the boundaries of the interval and subtracting the values: \( F_{X}(4) - F_{X}(2) \).These tasks will help in understanding the fundamental concepts of the Probability Density Function, Cumulative Distribution Function, and the application of these concepts to determine probabilities over specific intervals. ## Power LawsA random variable \( X \) follows the power law with parameter \( \rho > 0 \) when \( X \) has Probability Density Function (PDF) \( f_X(x) = c_{\rho} x^{-\rho} \) for \( x \ge 1 \), where \( \rho \) governs how fast the probabilities go to 0 as \( x \) goes to \( \infty \).

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Iculate cp so that Jx(x) is a vali Find CDF of X parameterized by p. Find P(2 < X < 4).

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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Question

### Problem Statement

To understand the proper formulation and properties of probability distributions, consider the following tasks:

1. **Calculate \( c_{\rho} \) so that \( f_{X}(x) \) is a valid Probability Density Function (PDF).**
- A valid PDF must satisfy the properties of being non-negative and integrating to 1 over its entire range.

2. **Find the Cumulative Distribution Function (CDF) of \( X \) parameterized by \( \rho \).**
- The CDF, \( F_{X}(x) \), represents the probability that the random variable \( X \) is less than or equal to a certain value \( x \) and is derived by integrating the PDF over the relevant range.

3. **Find \( P(2 < X \leq 4) \).**
- This probability can be found by evaluating the CDF at the boundaries of the interval and subtracting the values: \( F_{X}(4) - F_{X}(2) \).

These tasks will help in understanding the fundamental concepts of the Probability Density Function, Cumulative Distribution Function, and the application of these concepts to determine probabilities over specific intervals.

## Power Laws

A random variable \( X \) follows the power law with parameter \( \rho > 0 \) when \( X \) has Probability Density Function (PDF) \( f_X(x) = c_{\rho} x^{-\rho} \) for \( x \ge 1 \), where \( \rho \) governs how fast the probabilities go to 0 as \( x \) goes to \( \infty \).

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