A family of pdf's that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and ?, both > 0, and the pdf is as follows If the rv X has pdf f(x; k, ?), for any fixed b > ?, obtain an expression for P(X ≤ b).For ? < a < b, obtain an expression for the probability P(a ≤ X ≤ b). The image displays a probability density function commonly used in statistics. It is defined as follows:\[ f(x; k, \theta) = \begin{cases} \frac{k \cdot \theta^k}{x^{k + 1}} & \text{for } x \geq \theta \\ 0 & \text{for } x < \theta \end{cases} \]### Explanation- **\( f(x; k, \theta) \)**: Represents the probability density function with parameters \( k \) and \( \theta \).- **Case 1 (\( x \geq \theta \))**: The function is defined as \(\frac{k \cdot \theta^k}{x^{k + 1}}\). This part of the function applies when the variable \( x \) is greater than or equal to \( \theta \).- **Case 2 (\( x < \theta \))**: The function equals zero, indicating no probability mass is assigned to this region.This function is used primarily in statistical calculations involving distributions that have a threshold or minimum value, such as the Pareto distribution. It accounts for events where a condition or minimum threshold must be met for the event to have a non-zero probability.

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Answered: If the rv X has pdf f(x; k, ?), for any…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A family of pdf's that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and ?, both > 0, and the pdf is as follows

If the rv X has pdf f(x; k, ?), for any fixed b > ?, obtain an expression for P(X ≤ b).

For ? < a < b, obtain an expression for the probability P(a ≤ X ≤ b).

The image displays a probability density function commonly used in statistics. It is defined as follows:

\[
f(x; k, \theta) =
\begin{cases}
\frac{k \cdot \theta^k}{x^{k + 1}} & \text{for } x \geq \theta \\
0 & \text{for } x < \theta
\end{cases}
\]

### Explanation
- **\( f(x; k, \theta) \)**: Represents the probability density function with parameters \( k \) and \( \theta \).
- **Case 1 (\( x \geq \theta \))**: The function is defined as \(\frac{k \cdot \theta^k}{x^{k + 1}}\). This part of the function applies when the variable \( x \) is greater than or equal to \( \theta \).
- **Case 2 (\( x < \theta \))**: The function equals zero, indicating no probability mass is assigned to this region.

This function is used primarily in statistical calculations involving distributions that have a threshold or minimum value, such as the Pareto distribution. It accounts for events where a condition or minimum threshold must be met for the event to have a non-zero probability.

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

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