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Question #41009

Difficulty: 8/10 | 🟢 Free

The Question

A continuous random variable $X$ takes values in the interval $[0, \pi]$ . Its probability density function is given by $f(x) = K(F''(x) + F(x))$ , where $K$ is a positive constant and $F(x) = \int_{0}^{x} \sin(x-t) \arccos(\cos 2t) dt$ What is the variance of $X$ ?

Options

A.
$\frac{\pi^2}{24}$
B.
$\frac{\pi^2}{18}$
C.
$\frac{\pi^2}{12}$
D.
$\frac{\pi^2}{8}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Axiomatic Probability Constraints|Leibniz Rule for Differentiation under Integral|Principal Value Branch Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the Leibniz Rule with the principal value branches of inverse trigonometric functions.

Students must first recognize the differential identity $F''(x) + F(x) = g(x)$ generated by the integral.

The primary trap lies in blindly simplifying $\arccos(\cos 2x)$ to $2x$ over the entire interval $[0, \pi]$ .

This ignores the branch cut at $x = \pi/2$ , leading to an asymmetric, incorrect PDF $f(x) = 2Kx$ .

Consequently, the normalization constant $K$ and the variance are computed incorrectly (yielding $\pi^2/18$ ).

Correctly splitting the domain reveals a symmetric triangular distribution, leading to the true variance of $\pi^2/24$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Axiomatic Probability Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free