Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

The Pareto probability density function has the form $$ f(x)= \begin{cases}\frac{C M^{k}}{x^{k+1}} & \text { if } x \geq M \\ 0 & \text { if } x

Step-by-Step Solution

Answer

(a) \( C = k \). (b) Mean \( \mu = \frac{kM}{k-1} \) (finite for \( k > 1 \)). (c) Variance \( \sigma^2 = \frac{k M^2}{(k-2)(k-1)^2} \) (finite for \( k > 2 \)).

Step 1: Find C for normalization

For a function to be a probability density function, its integral over the entire space must be equal to 1. So we need to find \( C \) such that:\[ \int_{M}^{\infty} \frac{C M^{k}}{x^{k+1}} \, dx = 1 \]Calculate the integral:\[ \int_{M}^{\infty} \frac{M^{k}}{x^{k+1}} \ dx = M^{k} \int_{M}^{\infty} x^{-(k+1)} \, dx \]Solve the integral:\[ = M^{k} \left[ \frac{x^{-k}}{-k} \right]_{M}^{\infty} = \frac{M^{k}}{k} \left[ 0 - (M^{-k}) \right] = \frac{1}{k} \]Thus, we set \( C = k \) so that the whole is 1:\[ C \cdot \frac{1}{k} = 1 \Rightarrow C = k \]

Step 2: Calculate the Mean

Given \( C = k \), the mean \( \mu \) is computed by evaluating:\[ \mu = \int_{M}^{\infty} x \cdot \frac{k M^{k}}{x^{k+1}} \, dx \]Simplify and evaluate:\[ = k M^{k} \int_{M}^{\infty} x^{-k} \, dx \]Evaluating the integral:\[ = k M^{k} \left[ \frac{x^{-(k-1)}}{-(k-1)} \right]_{M}^{\infty} \]\[ = \frac{k M^{k}}{k-1} \cdot [ 0 - M^{-(k-1)} ] \]This gives:\[ \mu = \frac{k M}{k-1} \]The mean is finite only if \( k > 1 \). Otherwise, the integral diverges, indicating an infinite mean.

Step 3: Calculate the Variance

The variance \( \sigma^2 \) is given by:\[ \sigma^2 = \int_{M}^{\infty} (x - \mu)^2 \cdot \frac{k M^{k}}{x^{k+1}} \, dx \]This simplifies to:\[ = \int_{M}^{\infty} x^2 \cdot \frac{k M^{k}}{x^{k+1}} \, dx - \mu^2 \]Calculate the first integral:\[ \int_{M}^{\infty} x^{1-k} \, dx = \frac{x^{2-k}}{2-k} \bigg|_{M}^{\infty} \]This integral is finite only for \( k > 2 \), leading to:\[ \sigma^2 = \frac{k M^2}{(k-2)(k-1)^2} \]If \( k \leq 2 \), the variance is infinite as the integral diverges.

Key Concepts

Probability Density Function

The probability density function (PDF) is a fundamental concept in probability theory and statistics. It describes the likelihood of a random variable taking on a particular value. For any function to be considered as a legitimate probability density function, its integral over the entire space must equal 1.

In the case of the Pareto distribution, we have: \[ f(x)= \begin{cases}\frac{C M^{k}}{x^{k+1}} & \text { if } x \geq M \ 0 & \text { if } x

• The constant \(M\) serves as the minimum possible value of \(x\).
• \(k\) is the shape parameter affecting the distribution's tail.

To find \(C\), we calculate the integral of \(f(x)\) from \(M\) to infinity and set it to 1. This ensures the total probability adds up to 1.

Mean of a Probability Distribution

The mean of a probability distribution, also known as the expected value, provides a measure of the central tendency of the distribution.

For the Pareto distribution, the mean \(\mu\) is computed by: \[ \mu = \int_{M}^{\infty} x \cdot \frac{k M^{k}}{x^{k+1}} \, dx \] This simplifies to: \[ \mu = \frac{k M}{k-1} \] The mean is only finite when \(k > 1\).
This is because if \(k \leq 1\), the resulting integral diverges, leading to an infinite mean. It's crucial to check the conditions affecting the parameters as this will impact the convergence for calculations.

Variance Calculation

Variance is a measure of how much the values of a random variable spread out from the mean. It is a fundamental concept in probability theory used to understand data distribution more deeply.

The variance \(\sigma^2\) of the Pareto distribution is calculated using: \[ \sigma^2 = \int_{M}^{\infty} (x - \mu)^2 \cdot \frac{k M^{k}}{x^{k+1}} \, dx \] This leads to: \[ \sigma^2 = \frac{k M^2}{(k-2)(k-1)^2} \] The variance is finite only for \(k > 2\).
Otherwise, it becomes infinite, demonstrating why understanding the underlying conditions and parameters is important in probability calculations, especially for different ranges of \(k\).

Probability Theory

Probability theory is the mathematical framework for analyzing random phenomena and helps in the formulation of the laws of probability.

Key aspects include:

• Random Variables: These variables assume different outcomes and are described using probability distributions.
• Probability Distributions: Functions that describe the likelihood of each possible value that the random variable takes.

The Pareto distribution is a specific example that serves to model data sets with a heavy-tail distribution, typical in economic contexts.
Understanding the interplay between different parameters and their effects on mean and variance is crucial in applying probability theory to real-world problems.

Integrals in Calculus

Integrals are a core concept in calculus and are essential for computing quantities such as area under curves and the cumulative values of probability distributions.

In the context of probability density functions like the Pareto distribution, integraion is used to verify that the function describes a valid probability distribution:
\[ \int_{M}^{\infty} \frac{C M^{k}}{x^{k+1}} \, dx = 1 \] Integration helps in finding key properties such as the mean and variance, by solving the respective integral equations.
A strong grasp of integration techniques, particularly handling infinite limits and improper integrals, is vital in facilitating these calculations and fully understanding the behavior of probability distributions.