Math Problem Statement
The density of X is given by f(x) = { 10/x^2, for x > 10; 0, for x <= 10 }. What is the distribution of X? Find P(X > 20).
Solution
The problem provides the probability density function (PDF) for a random variable , defined as:
\frac{10}{x^2}, & \text{for } x > 10 \\ 0, & \text{for } x \leq 10 \end{cases}$$ You are asked two questions: 1. **What is the distribution of $$X$$?** 2. **Find $$P(X > 20)$$.** ### Step 1: Identifying the Distribution of $$X$$ The given probability density function resembles a **Pareto distribution** of the second kind, also called a Lomax distribution, with the following characteristics: - The PDF is proportional to $$\frac{1}{x^2}$$, which is typical of the tail of a Pareto distribution. - The support is $$x > 10$$, indicating a lower bound at $$x = 10$$. Thus, $$X$$ follows a Pareto distribution (Type II) with shape parameter $$\alpha = 2$$ and scale parameter $$x_m = 10$$. The PDF of a Pareto distribution with these parameters is given by: $$f(x) = \frac{\alpha x_m^\alpha}{x^{\alpha + 1}}, \quad \text{for } x > x_m$$ For this problem, $$\alpha = 2$$ and $$x_m = 10$$, so: $$f(x) = \frac{10}{x^2}, \quad \text{for } x > 10$$ ### Step 2: Finding $$P(X > 20)$$ The cumulative distribution function (CDF) for a Pareto distribution is given by: $$F(x) = 1 - \left( \frac{x_m}{x} \right)^\alpha, \quad \text{for } x > x_m$$ To find $$P(X > 20)$$, we calculate the complement of the CDF at $$x = 20$$: $$P(X > 20) = 1 - F(20) = \left( \frac{10}{20} \right)^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}$$ Thus, $$P(X > 20) = \frac{1}{4} = 0.25$$. ### Final Answers: 1. The distribution of $$X$$ is **Pareto (Type II)** with parameters $$\alpha = 2$$ and $$x_m = 10$$. 2. $$P(X > 20) = 0.25$$. Would you like further details or have any questions? Here are 5 related questions to deepen your understanding: 1. How would you derive the mean and variance of a Pareto distribution? 2. What is the significance of the shape parameter $$\alpha$$ in a Pareto distribution? 3. How would the probability change if we wanted to compute $$P(X > 30)$$? 4. Can we normalize the PDF given to ensure it integrates to 1? 5. How does a Type II Pareto distribution differ from the standard Pareto distribution? **Tip:** In any continuous distribution, the probability density function (PDF) must integrate to 1 over its domain. Always check this when dealing with a PDF!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Density Functions
Pareto Distribution
Formulas
f(x) = 10/x^2 for x > 10
CDF: F(x) = 1 - (xm/x)^α
P(X > 20) = 1 - F(20)
Theorems
Pareto Distribution
Suitable Grade Level
College/University Level (Probability and Statistics)
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