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Question #41449

Difficulty: 8/10 | 🟢 Free

The Question

A discrete random variable $X_n$ takes values $k \in \{1, 2, \dots, 2n\}$ . A transformed random variable is defined as $U_n = \frac{X_n - n}{n}$ . The probability mass function of $X_n$ is given by $P(X_n = k) = \frac{1}{Z_n} (u_k + 1)^2 \cos\left( \frac{\pi}{2} u_k \right)$ where $u_k = \frac{k - n}{n}$ and $Z_n$ is the normalization constant. Evaluate $\lim_{n \to \infty} E[U_n]$ .

Options

A.
$\frac{\pi^2 - 8}{\pi^2 - 4}$
B.
$\frac{\pi^2 - 4}{\pi^2 - 8}$
C.
$\frac{\pi^2 - 8}{\pi^2 + 4}$
D.
$\frac{2\pi^2 - 16}{\pi^2 - 4}$

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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

Integral as the Limit of a Sum|Probability Mass Function (PMF)|Step-Deviation Method (u = (x-a)/h)

Why This Question Is Tricky

Common Mistakes

This question bridges probability and calculus by requiring the evaluation of an expectation as the limit of a Riemann sum.

The core trap lies in the step-deviation substitution $u_k = (k-n)/n$ .

Students often struggle to correctly map the discrete summation limits ( $k=1$ to $2n$ ) to the continuous integral bounds ( $u=-1$ to $1$ ) and fail to recognize $\Delta u = 1/n$ as the differential $du$ .

A common mistake is expanding the sum in terms of $x = k/n$ and integrating from $0$ to $2$ , which bypasses the elegant cancellation of odd functions in the symmetric interval $[-1, 1]$ , leading to severe algebraic entanglement and a high probability of calculation errors.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integral as the Limit of a Sum
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free