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Question #41003

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x)$ be a continuous probability density function defined on $[0, \infty)$ . It is given that $y = f(x)$ satisfies the differential equation $\frac{dy}{dx} + y = \frac{1}{6}(3x^2 - 4x + 2) e^{-x}$ with the initial condition $f(0) = \frac{1}{3}$ . If $M$ is the mode of the distribution, find the mean absolute deviation of the distribution about $M$ .

Options

A.
$3$
B.
$39 e^{-3}$
C.
$22 e^{-2} - 1$
D.
$17 e^{-3}$

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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

Mean Deviation for Continuous Frequency Distribution|Linear Differential Equation Structure|Critical Point vs Stationary Point

Why This Question Is Tricky

Common Mistakes

This question synthesizes linear differential equations with continuous probability distributions, specifically targeting the conceptual difference between a stationary point and a critical point when finding the mode.

The mode is defined as the global maximum of the probability density function.

Students commonly fall into the trap of merely setting $f'(x) = 0$ to find stationary points, which yields a local maximum at $x=3$ .

Assuming $M=3$ leads to the incorrect mean deviation of $39 e^{-3}$ .

However, the true global maximum occurs at the domain boundary $x=0$ (a critical point where the derivative is not zero).

Recognizing $M=0$ simplifies the mean deviation to the distribution's mean, correctly yielding 3.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Mean Deviation for Continuous Frequency Distribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free