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Question #40992

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable with the probability mass function $P(X=x) = \frac{x}{15}$ for $x \in \{1, 2, 3, 4, 5\}$ . If $Y = |X - 3|$ , what is the ratio of the standard deviation of $Y$ to its expectation, $\frac{\sigma_Y}{E[Y]}$ ?

Options

A.
$\frac{\sqrt{14}}{6}$
B.
$\frac{\sqrt{14}}{2}$
C.
$\frac{5\sqrt{14}}{18}$
D.
$\frac{3\sqrt{14}}{10}$

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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 fundamental definition of expectation and variance under non-linear transformations, specifically the modulus function.

The 'Lethal Collision' occurs when students incorrectly assume $E[|X-k|] = |E[X]-k|$ or $Var(|X-k|) = Var(X-k) = Var(X)$ .

Because the modulus folds the distribution around $X=3$ , the summation must be split piecewise to find the true distribution of $Y$ .

Failing to do so leads to calculating the variance of $X$ instead of $Y$ , yielding $14/9$ instead of $14/25$ , which cascades into an incorrect coefficient of variation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Standard Deviation and Coefficient of Variation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free