MMiDS 1.4: Self-Assessment Quiz

The volume of the \(d\)-dimensional cube \(C = [-1/2, 1/2]^d\) is:

• a) \(1/d\)
• b) \(1/2^d\)
• c) 1
• d) \(2^d\)

In a high-dimensional cube \(C = [-1/2, 1/2]^d\), as the dimension \(d\) increases, the probability that a randomly chosen point lies within the inscribed sphere \(B = \{x \in \mathbb{R}^d : \|x\| \le 1/2\}\):

• a) Approaches 1
• b) Approaches 1/2
• c) Approaches 0
• d) Remains constant

Which of the following best describes the appearance of a high-dimensional cube?

• a) A smooth, round ball
• b) A spiky ball with most of its volume concentrated in the corners
• c) A perfect sphere with uniform volume distribution
• d) A flat, pancake-like shape

In high dimensions, a standard Normal \(d\)-vector \(X\) is most likely to have a norm \(\|X\|\) close to:

• a) 0
• b) 1
• c) \(\sqrt{d}\)
• d) \(d\)

In the context of high-dimensional Gaussians, which of the following statements is true?

• a) The joint PDF is maximized at the origin for all dimensions.
• b) There are many ways to obtain a norm of 0.
• c) There is only one way to obtain a norm of 0.
• d) The joint PDF is minimized at the origin for all dimensions.

In the proof of the theorem about high-dimensional Gaussians, which property of the squared norm \(\|X\|^2\) is used?

• a) It is a sum of dependent random variables.
• b) It is a sum of independent random variables.
• c) It is a product of independent random variables.
• d) It is a product of dependent random variables.

Which inequality is used to prove the theorems about high-dimensional cubes and Gaussians?

• a) Cauchy-Schwarz inequality
• b) Triangle inequality
• c) Markov's inequality
• d) Chebyshev's inequality