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Question #59489

Difficulty: 5/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable taking values in the set $\{1, 2, 3, 4, 5\}$ with probability distribution $P(X=x) = \frac{x}{15}$ . If a new random variable is defined as $Y = |X - 3|$ , what is the coefficient of variation (defined as the ratio $\frac{\sigma_Y}{\mu_Y}$ ) of $Y$ ?

Options

A.
$\frac{\sqrt{14}}{6}$
B.
$\frac{\sqrt{14}}{2}$
C.
$\frac{5\sqrt{14}}{18}$
D.
$\frac{\sqrt{14}}{3}$

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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

Standard Deviation and Coefficient of Variation|Random Variable Expectation|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the calculation of the Coefficient of Variation for a transformed discrete random variable involving a modulus function.

The primary trap is the 'Modulus Trap' in expectation: students often incorrectly assume that $E[|X-k|] = |E[X]-k|$ .

Here, $E[X] = 11/3$ , so $|E[X]-3| = 2/3$ , which drastically differs from the true piecewise expectation $E[Y] = 6/5$ .

A secondary trap is assuming $Var(|X-k|) = Var(X-k) = Var(X)$ .

While $E[|X-k|^2] = E[(X-k)^2]$ , the variance $Var(|X-k|)$ subtracts $(E[|X-k|])^2$ , not $(E[X-k])^2$ .

Falling for these traps leads to incorrect variance and expectation values, invalidating the final ratio.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Standard Deviation and Coefficient of Variation
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free