A manufacturer’s annual losses follow a distribution with density function
f(x)={\matrix((2.5(0.6)^2.5)/x^3.5 ,&x>0.6@0,&otherwise)┤
Calculate E[min⁡(X-2,0)]

Use integration methods and return the final number

We are tasked with calculating \( E[\min(X - 2, 0)] \) where \( X \) follows a distribution with density function:

\[
f(x) = \begin{cases} 
\frac{2.5(0.6)^{2.5}}{x^{3.5}}, & x > 0.6 \\
0, & \text{otherwise}
\end{cases}
\]

The function \( \min(X - 2, 0) \) can be broken down as:

\[
\min(X - 2, 0) = \begin{cases} 
0, & \text{if } X \geq 2 \\
X - 2, & \text{if } X < 2 
\end{cases}
\]

So, the expected value can be split into two parts:

1. For \( X \geq 2 \), \( \min(X - 2, 0) = 0 \).
2. For \( X < 2 \), \( \min(X - 2, 0) = X - 2 \).

Thus, we can express the expectation as:

\[
E[\min(X - 2, 0)] = \int_{0.6}^{2} (x - 2) f(x) dx
\]

Now, let's compute this integral step-by-step. The density function in the range \( [0.6, 2] \) is given by \( f(x) = \frac{2.5(0.6)^{2.5}}{x^{3.5}} \).

### Setting up the integral:

\[
E[\min(X - 2, 0)] = \int_{0.6}^{2} (x - 2) \cdot \frac{2.5(0.6)^{2.5}}{x^{3.5}} dx
\]

I will now calculate this integral.The expected value \( E[\min(X - 2, 0)] \) is approximately \( -1.0657 \).

Would you like more details or have any questions? 

Here are 5 related questions to consider:

1. How does the density function \( f(x) \) affect the expected value?
2. What would happen if the threshold in the min function was changed from 2 to another value?
3. How do integration limits affect the result in expectation problems like this?
4. Can this expectation be approximated using numerical methods other than integration?
5. What is the interpretation of a negative expected value in this context?

**Tip**: When calculating expected values, splitting the range of the function into manageable parts, as we did here, can simplify complex expressions.

A manufacturer’s annual losses follow a distribution with density function f(x)={\matrix((2.5(0.6)^2.5)/x^3.5 ,&x>0.6@0,&otherwise)┤. Calculate E[min⁡(X-2,0)] using integration methods and return the final number.

Learn how to compute the expected value E[min(X - 2, 0)] for a given distribution density function using integration. This detailed step-by-step solution explains key probability concepts and theorems, making it ideal for undergraduate-level students studying probability or statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation