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Question #41844

Difficulty: 8/10 | 🟢 Free

The Question

A discrete random variable $Y_n$ takes values $k \in \{1, 2, \dots, n\}$ with probabilities given by the probability mass function: $P(Y_n = k) = \frac{1}{\lambda_n} \left( \frac{\pi}{2n} \sin\left(\frac{\pi k}{2n}\right) + \frac{1}{k^2+k} \right)$ where $\lambda_n$ is the normalization constant such that $\sum_{k=1}^n P(Y_n = k) = 1$ . Let $X_n = \frac{Y_n}{n}$ . Find the value of $\lim_{n \to \infty} \text{Var}(X_n)$ .

Options

A.
$\frac{2\pi - 5}{\pi^2}$
B.
$\frac{2\pi - 4}{\pi^2}$
C.
$\frac{\pi - 2}{\pi^2}$
D.
$\frac{4\pi - 12}{\pi^2}$

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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

Variance of Discrete Variable|Method of Differences|Integral as the Limit of a Sum

Why This Question Is Tricky

Common Mistakes

This question engineers a 'lethal collision' between the Method of Differences and the Integral as the Limit of a Sum within a probability context.

The student must normalize a probability mass function that is a linear combination of a telescoping series and a Riemann sum sequence.

The trap lies in the calculation of the moments $E[X_n]$ and $E[X_n^2]$ : the $1/n$ and $1/n^2$ factors introduced by the variable transformation $X_n = Y_n/n$ perfectly convert the trigonometric terms into definite integrals while simultaneously driving the telescoping contributions to zero.

Students who attempt to evaluate the trigonometric sum algebraically or fail to separate the discrete and continuous limits will be caught in an algebraic nightmare.

Quick Facts About This Question

Question Information

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