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We are tasked with finding the expected value \( \mathbb{E}(X) \) for a continuous random variable \( X \), whose probability density function \( f(x) \) is given by:

\[
f(x) = \begin{cases} 
\frac{\pi}{12} \cos\left(\frac{\pi x}{12}\right), & x \in [0, 6] \\
0, & \text{otherwise}
\end{cases}
\]

The expected value \( \mathbb{E}(X) \) for a continuous random variable is defined as:

\[
\mathbb{E}(X) = \int_{-\infty}^{\infty} x f(x) \, dx
\]

Since \( f(x) = 0 \) for \( x \notin [0, 6] \), the integral reduces to:

\[
\mathbb{E}(X) = \int_0^6 x \cdot \frac{\pi}{12} \cos\left(\frac{\pi x}{12}\right) \, dx
\]

We are also given the derivative of \( x \sin\left(\frac{\pi x}{12}\right) \):

\[
\frac{d}{dx} \left( x \sin\left(\frac{\pi x}{12}\right) \right) = \frac{\pi x}{12} \cos\left(\frac{\pi x}{12}\right) + \sin\left(\frac{\pi x}{12}\right)
\]

Let's use this information to compute \( \mathbb{E}(X) \).The expected value \( \mathbb{E}(X) \) is:

\[
\mathbb{E}(X) = 6 - \frac{12}{\pi}
\]

This is the simplified form of the expected value for the given probability density function.

Would you like further details on this computation or have any other questions?

Here are 5 related questions to expand on this topic:
1. How is the expected value \( \mathbb{E}(X) \) interpreted in probability theory?
2. What is the significance of the given derivative in this problem?
3. How would we compute the variance \( \text{Var}(X) \) for this random variable?
4. Can we generalize this approach for different probability density functions?
5. How would the result change if the range of \( x \) were different?

**Tip:** The expected value of a continuous random variable gives the long-term average or the center of the probability distribution.

A continuous random variable, X, has a probability density function. Given that the derivative of a specific function involving X is also provided, find E(X).

Explore how to compute the expected value E(X) for a continuous random variable whose probability density function involves trigonometric functions. This detailed guide utilizes calculus concepts like integration and derivative identities.

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