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Question #41838

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a continuous random variable taking values in the interval $[0, 1]$ . The probability density function of $X$ is given by $f(x) = k \\frac{d}{dx} \\left( \\int_{x}^{2x} (|t - 1| + t) dt \\right)$ where $k$ is a positive constant. If $M$ is the median of the distribution, find the value of $M$ .

Options

A.
$\\frac{3}{4}$
B.
$\\frac{1}{2}$
C.
$\\frac{5}{8}$
D.
$\\frac{2}{3}$

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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

Equiprobable Outcomes|Leibniz Rule for Differentiation under Integral|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Leibniz rule for differentiation under the integral sign combined with piecewise modulus analysis.

The primary trap lies in the upper limit of the integral, $2x$ .

Since $x \\in [0, 1]$ , students often incorrectly assume that the dummy variable $t$ remains $\\le 1$ throughout the integration, leading them to simplify $|t - 1|$ as $1 - t$ everywhere.

This flawed assumption results in a constant derivative $g'(x) = 1$ , falsely implying that $X$ follows a uniform (equiprobable) distribution on $[0, 1]$ and yielding an incorrect median of $1/2$ .

A rigorous solution requires recognizing that $2x$ crosses $1$ at $x = 1/2$ , necessitating a piecewise breakdown of the PDF before calculating the normalization constant and the median.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Equiprobable Outcomes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free