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Question #41817

Difficulty: 8/10 | 🟢 Free

The Question

Let $Y$ be a random variable chosen uniformly from the set $\{-2, -1, 0, 1, 2\}$ . A set of 5 distinct objects is divided into $k$ non-empty subsets, where $k = 3 + \text{sgn}(Y)$ . All valid partitions of the 5 objects into $k$ subsets are equally likely. Let $X$ be the random variable representing the number of objects in the subset containing a specific object $O_1$ . Find the expected value $E[X]$ .

Options

A.
$\frac{787}{375}$
B.
$\frac{128}{75}$
C.
$\frac{107}{30}$
D.
$\frac{51}{25}$

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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

Random Variable Expectation|Division into Groups|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of discrete probability, Stirling numbers of the second kind, and the precise definition of the signum function.

The 'Lethal Collision' occurs because the number of groups $k$ depends on $\text{sgn}(Y)$ .

Students frequently assume $\text{sgn}(Y)$ only takes values $\pm 1$ , completely overlooking the point discontinuity at $Y=0$ where $\text{sgn}(0)=0$ .

This leads to a loss of probability mass (ignoring the $k=3$ case).

Furthermore, calculating the expected group size requires understanding that the probability of a specific item belonging to a group of size $m$ is proportional to $m$ , meaning the expected size is related to the sum of squares of the partition sizes, not just a simple average.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Random Variable Expectation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free