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The problem provides the probability density function (PDF) for a random variable \( X \), defined as:

\[
f(x) = \begin{cases} 
\frac{10}{x^2}, & \text{for } x > 10 \\
0, & \text{for } x \leq 10 
\end{cases}
\]

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!

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).

Learn how to identify the distribution of X with the given PDF and calculate the probability P(X > 20). This problem involves the Pareto distribution (Type II) with a detailed step-by-step solution, suitable for students studying probability and statistics at the college level.

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