If 
𝑋
X is a random variable with probability density function 
𝑓
(
𝑥
)
f(x), then the expected value of 
𝑋
3
X 
3
  is given by:

A. 
𝐸
(
𝑋
3
)
=
∫
𝑥
3
/
2
𝑓
(
𝑥
)
𝑑
𝑥
E(X 
3
 )=∫x 
3/2
 f(x)dx
B. 
𝐸
(
𝑋
3
)
=
∫
𝑥
3
𝑓
(
𝑥
)
𝑑
𝑥
E(X 
3
 )=∫x 
3
 f(x)dx
C. 
𝐸
(
𝑋
3
)
=
∫
𝑥
6
𝑓
(
𝑥
)
𝑑
𝑥
E(X 
3
 )=∫x 
6
 f(x)dx
D. 
𝐸
(
𝑋
3
)
=
∫
𝑥
4
𝑓
(
𝑥
)
𝑑
𝑥
E(X 
3
 )=∫x 
4
 f(x)dx

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.

If X is a random variable with probability density function f(x), then the expected value of X^3 is given by:

A. E(X^3) = ∫x^(3/2) f(x) dx
B. E(X^3) = ∫x^3 f(x) dx
C. E(X^3) = ∫x^6 f(x) dx
D. E(X^3) = ∫x^4 f(x) dx

Understand how to calculate the expected value of X^3 for a random variable with a probability density function. Explore the formula E(X^n) = ∫x^n f(x) dx, key concepts like integration, and the definition of expected value. Ideal for undergraduate students studying probability and statistics.

Math Problem Statement

A. 𝐸 ( 𝑋 3 )

∫ 𝑥 3 / 2 𝑓 ( 𝑥 ) 𝑑 𝑥 E(X 3 )=∫x 3/2 f(x)dx B. 𝐸 ( 𝑋 3 )

∫ 𝑥 3 𝑓 ( 𝑥 ) 𝑑 𝑥 E(X 3 )=∫x 3 f(x)dx C. 𝐸 ( 𝑋 3 )

∫ 𝑥 6 𝑓 ( 𝑥 ) 𝑑 𝑥 E(X 3 )=∫x 6 f(x)dx D. 𝐸 ( 𝑋 3 )

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