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Question #41847

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable taking values $r \in \{0, 1, 2, \dots, 153\}$ . The probability distribution of $X$ is given by $P(X=r) = \lim_{m \to \infty} \text{Coefficient of } x^r \text{ in } \left( \sum_{k=1}^m \frac{1}{\sqrt{9m^2 + k}} + x \sum_{j=1}^{2m} \frac{1}{\sqrt{9m^2 + j}} \right)^{153}$ Find the value of $E[X] - V(X)$ , where $E[X]$ and $V(X)$ denote the expectation and variance of $X$ respectively.

Options

A.
68
B.
17
C.
34
D.
136

Check Your Answer

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

Variance of Discrete Variable|Coefficient of x^r in Expansion|Squeeze Theorem (Sandwich Theorem)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of limit evaluation via the Squeeze Theorem with the identification of discrete probability distributions.

A common trap is attempting to evaluate the sums as definite integrals (Riemann sums), which fails because the index $k$ is not scaled by $m^2$ in the differential limit.

Students must recognize that the bounds of the sums converge to constants $1/3$ and $2/3$ .

Furthermore, the '153' trap acts as a decoy; students might assume a Poisson distribution or misidentify the success probability $p$ , leading to the reversed symmetry trap of $E[X] - V(X) = 17$ instead of the correct $68$ .

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