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Question #41843

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable taking values in $\{0, 1, 2, \dots, 8\}$ . The probability distribution of $X$ is given by $P(X=k) = \frac{a_k}{3^4}$ , where $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)^4$ . A new random variable is defined as $Y = \left[ \frac{X}{3} \right]$ , where $[ \cdot ]$ denotes the greatest integer function. Find the variance of $Y$ .

Options

A.
$\frac{10}{27}$
B.
$\frac{8}{27}$
C.
$\frac{11}{27}$
D.
$\frac{4}{9}$

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|Multinomial Theorem|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle discrete random variables derived from multinomial expansions, combined with the non-linear Greatest Integer Function (step function).

The primary trap lies in assuming linearity of variance, i.e., incorrectly calculating $\text{Var}(X/3) = \frac{1}{9}\text{Var}(X) = \frac{8}{27}$ .

The step function breaks the linearity of the original distribution, forcing the student to manually partition the sample space and compute the moments $E[Y]$ and $E[Y^2]$ from first principles.

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