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Question #59939

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{x} = (x_1, x_2, \dots, x_n)^T \in \mathbb{R}^n$ represent a dataset of $n$ observations with mean $\mu \neq 0$ and variance $\sigma^2 \neq 0$ . Let $\vec{e} = (1, 1, \dots, 1)^T \in \mathbb{R}^n$ . A new dataset is represented by the vector $\vec{y} = \vec{e} - 2 \left( \frac{\vec{e} \cdot \vec{x}}{\|\vec{x}\|^2} \right) \vec{x}$ . Which of the following represents the variance of the dataset $\vec{y}$ ?

Options

A.
$\frac{4\mu^2 \sigma^2}{(\sigma^2+\mu^2)^2}$
B.
1
C.
$\frac{4\mu^2 \sigma^2}{\sigma^2+\mu^2}$
D.
$\frac{(\sigma^2-\mu^2)^2}{(\sigma^2+\mu^2)^2}$

Check Your Answer

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

Variance Invariance under Translation|Linear Independence vs Dependence|Dot Product Projection Sign

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the geometric interpretation of variance and the properties of translation and scaling in vector spaces.

The specific trap is the 'Second Moment Trap' via a Householder reflection.

Because $\vec{y}$ is a reflection of $\vec{e}$ across the orthogonal complement of $\vec{x}$ , its Euclidean norm is preserved ( $\left\|\vec{y}\right\|^2 = \left\|\vec{e}\right\|^2 = n$ ).

Students who blindly calculate the squared magnitude $\frac{1}{n}\left\|\vec{y}\right\|^2$ will get $1$ (Option B) and incorrectly assume this is the variance, forgetting that variance is the central moment.

To find the true variance, one must recognize that the $\vec{e}$ term is a pure translation that does not affect variance, allowing direct scaling of $\sigma^2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Variance Invariance under Translation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free