Consider the two statements. S_1: There exist random variables X and Y such that ≤ft(E≤ft[(X - E(X))\,(Y - E(Y))])^2 > Var[X]\,Var[Y] S_2: For all random variables X and Y, Cov[X, Y] = E≤ft[\,|X - E[X]|\;|Y - E[Y]|\,] Which one of the following choices is correct?

GATE CSE 2021 Set 1 Q45: Covariance bound and covariance with absolute deviations

Engineering Mathematics
2 marks
MCQ
−2/3 if wrong

Consider the two statements.

$S_1$ : There exist random variables $X$ and $Y$ such that

$\left(\mathbb{E}\left[(X - \mathbb{E}(X))\,(Y - \mathbb{E}(Y))\right]\right)^2 > \operatorname{Var}[X]\,\operatorname{Var}[Y]$

$S_2$ : For all random variables $X$ and $Y$ ,

$\operatorname{Cov}[X, Y] = \mathbb{E}\left[\,|X - \mathbb{E}[X]|\;|Y - \mathbb{E}[Y]|\,\right]$

Which one of the following choices is correct?

Try it: multiple choice: exactly one correct option

Wrong answer costs 2/3 mark in GATE.

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Answer: D (official GATE 2021 final answer key)

Concept used: Cauchy–Schwarz gives $\operatorname{Cov}[X,Y]^2 \le \operatorname{Var}[X]\operatorname{Var}[Y]$ ; covariance can be negative, an expectation of absolute values cannot.

First step: $S_1$ claims $\operatorname{Cov}^2 > \operatorname{Var}[X]\operatorname{Var}[Y]$ ; for $S_2$ try $Y = -X$ .

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key

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GATE CSE 2021 Set 1 Q45: Covariance bound and covariance with absolute deviations

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