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Question #41891

Difficulty: 8/10 | 🟢 Free

The Question

Let $\mathbf{1}_A$ denote the indicator function of an event $A$ . An integer $k$ is chosen uniformly at random from the set $S = \{1, 2, 3, \dots, 100\}$ . A random variable $Y$ is defined as $Y = \sum_{n=1}^{10} (2n+1) \cdot \mathbf{1}_{ \sqrt{k} - \{ \sqrt{k} \} = n }$ where $\{x\}$ denotes the fractional part of $x$ . Find the value of $100 \cdot E[Y]$ , where $E[Y]$ is the expected value of $Y$ .

Options

A.
1350
B.
1770
C.
1329
D.
1449

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

Indicator Function Logic|Sigma Notation Properties|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from an indicator function involving fractional parts to summation bounds.

The core trap lies in the boundary condition of the sample space.

Students frequently assume the algebraic inequality $n^2 \le k < (n+1)^2$ holds fully for all $n$ in the summation, blindly substituting $c_n = 2n+1$ .

This leads to the incorrect inclusion of out-of-bound values for $n=10$ (calculating 21 terms instead of 1), yielding the trap answer 1770.

It rigorously evaluates attention to domain constraints when floor functions partition a sequence.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Indicator Function Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free