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Question #41530

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a continuous random variable with probability density function $p(x)$ defined on $[1, \infty)$ as $p(x) = -c \left( f(x) + x f'(x) \right)$ where $f(x) = \frac{\ln(1+x)}{x^3}$ and $c$ is a positive constant. Find the expected value of $X$ , denoted by $E[X]$ .

Options

A.
2
B.
3
C.
$2 \ln 2$
D.
$\frac{3}{\ln 2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Integrating f(x) + xf'(x) forms|Random Variable Expectation|Probability Mass Function (PMF)

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of probability density normalization and advanced calculus.

The core trap lies in the improper integral evaluation at infinity.

Students often blindly integrate $f(x) + xf'(x)$ without recognizing it as the exact derivative $\frac{d}{dx}(xf(x))$ , leading to algebraic chaos.

Furthermore, the 'Lethal Collision' occurs if the student fails to rigorously apply L'Hôpital's rule for the convergence criteria at the upper limit ( $x \to \infty$ ), which is required twice (once for normalization, once for expectation).

Ignoring these boundary limits results in an undefined or incorrect constant $c$ , collapsing the entire expectation calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integrating f(x) + xf'(x) forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free