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Question #41005

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a continuous random variable with probability density function $f(x) = k(x^2 + |x|)$ for $x \in [-3, 3]$ and $f(x) = 0$ otherwise. Let $E$ be the event that the random variable $X$ satisfies the inequality: $\int_{-X}^{X} \left( t^3 \cos t + t^2 \sin^3 t + |t| - 1 \right) dt > 0$ Find the probability of event $E$ , $P(E)$ .

Options

A.
$\frac{1}{2}$
B.
$\frac{67}{81}$
C.
$\frac{53}{81}$
D.
$\frac{19}{36}$

Check Your Answer

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

Sample Space Construction|Sample Space Partitioning|Symmetry Utilization|Even and Odd Function Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to construct and partition a sample space using the properties of definite integrals, specifically even and odd function symmetry.

The primary trap lies in evaluating the integral with variable limits $[-X, X]$ .

Students often incorrectly assume the integral evaluates to $X^2 - 2X$ or $|X|^2 - 2|X|$ for all $X$ , ignoring the sign change introduced by the upper and lower limits when $X < 0$ .

This failure to utilize parity correctly leads to an improper partitioning of the sample space (e.g., yielding $X \in [-3, 0) \cup (2, 3]$ ), which cascades into incorrect integration limits for the probability density function and results in the common distractor 67/81.

Quick Facts About This Question

Question Information

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