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Question #40977

Difficulty: 8/10 | 🟢 Free

The Question

Let a discrete random variable $X$ take values from the set $S = \{0, 1, \frac{3}{2}, 2, 3\}$ . The probability mass function of $X$ is given by $P(X=x) = \frac{2x+2}{25}$ for all $x \in S$ . A new random variable is defined as $Y = \text{sgn}(X^2 - 3X + 2)$ , where $\text{sgn}(\cdot)$ denotes the signum function. If $\alpha$ is the mean deviation of $Y$ about its mean, and $\beta$ is the mean deviation of $Y$ about its median, find the value of $100(\alpha - \beta)$ .

Options

A.
2
B.
4
C.
5
D.
8

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

Mean Deviation about Mean vs. Median|Probability Mass Function (PMF)|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle transformations of discrete random variables using discontinuous functions (the signum function) and their understanding of statistical measures of dispersion.

The primary trap lies in the signum function's evaluation at the roots of the quadratic argument; students often overlook the $Y=0$ case, incorrectly assuming the signum function only outputs $\pm 1$ .

Furthermore, the problem forces the explicit calculation of both the mean deviation about the mean and the median.

A common conceptual error is conflating these two measures or failing to recognize that the median minimizes the mean absolute deviation, which serves as a theoretical check ( $\alpha > \beta$ ).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Mean Deviation about Mean vs. Median
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free