The expected value of $X^3$ for a random variable $X$ with probability density function $f(x)$ is calculated using the definition of expected value:

$E(X^3) = \int x^3 f(x) \, dx$

This means we multiply the function $x^3$ by the probability density function $f(x)$, and integrate over the range where $f(x)$ is defined.

Thus, the correct answer is:

B. $E(X^3) = \int x^3 f(x) \, dx$

Let me know if you want a detailed breakdown or further questions!

5 Related Questions:

1. What is the general formula for the expected value of $X^n$ for any $n$?
2. How does the concept of expected value extend to discrete random variables?
3. Can the expected value of $X^3$ be negative? Under what conditions?
4. What is the difference between the expected value and the variance of a random variable?
5. How do we calculate the expected value if the probability density function $f(x)$ is piecewise?

Tip:

Always check that the probability density function $f(x)$ satisfies the properties of being non-negative and integrates to 1 over its domain.