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Question #40995

Difficulty: 8/10 | 🟢 Free

The Question

A real number $x$ is chosen uniformly at random from the continuous interval $[0, 7]$ . Let the random variable $Y$ be defined as $Y = 10 + \int_0^{[x]} (2t - 3) dt$ where $[ \cdot ]$ denotes the greatest integer function. What is the coefficient of variation of $Y$ ?

Options

A.
$\frac{2\sqrt{3}}{7}$
B.
$\frac{\sqrt{14}}{7}$
C.
$\frac{2}{7}$
D.
$\frac{4\sqrt{3}}{7}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Standard Deviation and Coefficient of Variation|Equiprobable Outcomes|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between continuous and discrete random variables when step functions are involved.

The trap lies in the integral representation of the function, which baits students into applying the Fundamental Theorem of Calculus and treating $Y$ as a continuous variable $x^2 - 3x + 10$ over $[0, 7]$ .

However, because the upper limit of the integral is the Greatest Integer Function $[x]$ , the outcomes are actually a discrete set of equiprobable jumps.

Attempting to find the variance via continuous integration results in a 'Lethal Collision' with the true discrete probability mass function.

The correct approach requires evaluating the discrete states and computing the population variance algebraically.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Standard Deviation and Coefficient of Variation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free