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Question #41710

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_0^{\pi} x \sin^6 x \cos^4 x \, dx$ . A continuous random variable $X$ has the probability density function $f(x) = \begin{cases} \frac{512}{3\pi^2 \ln 3} \cdot \frac{I}{x^2+2x} & \text{for } x \ge \lambda \\ 0 & \text{otherwise} \end{cases}$ where $\lambda > 0$ is a constant such that $f(x)$ is a valid probability density function. Find the conditional probability $P\left(X > 2 \mid X > \frac{1}{2}\right)$ .

Options

A.
$\log_5 2$
B.
$\log_2 5$
C.
$\log_3 2$
D.
$\frac{1}{2} \log_5 2$

Check Your Answer

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

Conditional Probability Constraints|Definite Integral Properties (King's Property)|Bounded vs Unbounded Regions|Wallis' Formula

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of King's Property, Wallis' formula, and improper integrals within a probability context.

The primary trap lies in the convergence of the improper integral defining the total probability.

Students often blindly assume the lower bound of such PDFs is 0 (a common default), which here leads to $\int_0^\infty \frac{1}{x(x+2)} dx$ , a divergent integral that renders the probability space undefined.

By forcing the calculation of the normalization bound $\lambda$ , the problem ensures students verify the boundedness and convergence of the region.

Furthermore, the conditional probability constraint over an unbounded region ( $X > 1/2$ ) requires careful evaluation of the limit at infinity, penalizing those who rely on finite-bound shortcut formulas.

Quick Facts About This Question

Question Information

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