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Question #41778

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable taking values in a set $S$ . The set $S$ consists of all integers $x$ for which there exists at least one real number $y$ satisfying the system of inequalities: $y \ge 0, \quad x - y \ge 0, \quad x + 2y \le 4$ The probability mass function of $X$ is given by $P(X=x) = \begin{cases} k \frac{x+1}{2^x} & \text{for } x \in S \\ 0 & \text{otherwise} \end{cases}$ where $k$ is a positive constant. The expected value $E[X]$ is:

Options

A.
$\frac{28}{19}$
B.
2
C.
$\frac{16}{13}$
D.
$\frac{84}{41}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Non-Negativity Restriction|Probability Mass Function (PMF)|Random Variable Expectation

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to correctly identify the finite support of a random variable bounded by a system of linear inequalities (LP).

The non-negativity constraint $y \ge 0$ combined with $x \ge y$ forces $x \ge 0$ , while $x + 2y \le 4$ forces the upper bound $x \le 4$ .

The primary trap is the mechanical application of the infinite arithmetico-geometric series formula for expectation.

Students who ignore the LP boundaries will sum the PMF to infinity, incorrectly obtaining $E[X] = 2$ .

Other options represent partial boundary failures, such as excluding $x=0$ or miscalculating the upper bound as $x=3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Non-Negativity Restriction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free