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Question #40659

Difficulty: 8/10 | 🟢 Free

The Question

Let $Z$ be a random variable taking values in the set $\{0, 1, \dots, n\}$ . The probability generating function of $Z$ , defined as $G(t) = \sum_{k=0}^n P(Z=k) t^k$ , is given by the relation: $G(t) = \frac{\text{Coefficient of } x^{2n} \text{ in } (1+tx)^n (1+x)^{2n}}{\binom{3n}{2n}}$ If $W = 3Z - 2n$ , which of the following expressions represents the variance of $W$ ?

Options

A.
$\frac{4n^2}{3n-1}$
B.
$4n$
C.
$\frac{4n^2}{3n+1}$
D.
$\frac{2n^2}{3n-1}$

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

Expansion of Products of Binomials|Bernoulli Trials Logic|Random Variable Expectation|Symmetry and Invariance

Why This Question Is Tricky

Common Mistakes

This question tests the ability to extract moments from a Probability Generating Function (PGF) defined via a binomial product expansion.

The 'Lethal Collision' occurs when calculating the variance: students often mistakenly assume the underlying trials are independent (Binomial) and compute $Var(Z) = 4n/9$ , leading to the trap option $4n$ .

However, the PGF actually encodes a Hypergeometric distribution (dependent trials without replacement).

To find the true variance, students must rigorously apply the PGF moment formulas $E[Z]=G'(1)$ and $Var(Z)=G''(1)+G'(1)-(G'(1))^2$ , forcing them to manipulate binomial coefficients and correctly account for the covariance between non-i.i.d. trials implicitly captured by the polynomial expansion.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Expansion of Products of Binomials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free