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Question #41062

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{x^4}{12} - \frac{x^2}{2}$ . A continuous random variable $X$ has a probability density function $f_X(x)$ defined on $[-1, 1]$ which is directly proportional to $|f''(x)|$ . A new random variable is defined by the transformation $Y = X^3 - 3X$ . Find the probability that the roots of the quadratic equation $t^2 - Yt + \frac{3}{4} = 0$ are real.

Options

A.
$1 - \frac{\sqrt{3}}{2}$
B.
$1 - 2\sin\left(\frac{\pi}{9}\right)$
C.
$1 - \frac{\sqrt{2}}{2}$
D.
$\frac{1}{2}$

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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

Equiprobability Fallacy|Random Variable Mapping|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

This question tests the 'Equiprobability Fallacy' and the limitations of the Second Derivative Test in random variable mappings.

Students often assume that a non-linear transformation like $Y = X^3 - 3X$ will cause the probability density to concentrate at its local extrema (where $Y' = 0$ ), leading to asymptotes in the PDF of $Y$ .

However, the trap lies in the base PDF of $X$ , which is derived from the curvature $f''(x)$ .

Because $f_X(x)$ approaches zero at the exact same rate as the Jacobian $|dY/dX|$ at the critical points $X = \pm 1$ , the singularity is perfectly canceled out, resulting a uniform distribution for $Y$ .

Students who fall for the equiprobability fallacy by assuming $X$ is uniform, or who blindly apply the second derivative test expecting density concentration, will incorrectly calculate the probability using complex trigonometric roots.

Quick Facts About This Question

Question Information

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