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Question #41763

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable taking values $-1, 0,$ and $1$ with probabilities $p_1, p_2,$ and $p_3$ respectively. It is given that $p_2 \ge \frac{1}{5}$ and $|E[X]| \le \frac{2}{5}$ , where $E[X]$ denotes the expected value of $X$ . The maximum possible variance of $X$ is

Options

A.
$\frac{4}{5}$
B.
$\frac{16}{25}$
C.
$\frac{21}{25}$
D.
1

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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

Boundary Value Theorem|Variance of Discrete Variable|Calculation of Variance for Grouped Data

Why This Question Is Tricky

Common Mistakes

This problem tests the optimization of a non-linear objective function (Variance) over a linear feasible region defined by probability constraints.

Students accustomed to Linear Programming Problems (LPP) will instinctively evaluate the objective function only at the vertices of the feasible region.

The vertices of this region (e.g., $p_1=0.2, p_3=0.6$ ) yield a maximum variance of $16/25$ .

However, because the variance function is concave, its maximum can occur on an edge or in the interior.

The true maximum occurs at the midpoint of an edge where $p_1=p_3=0.4$ , yielding $4/5$ .

This highlights the danger of blindly applying the vertex theorem to non-linear objective functions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Boundary Value Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free