GATE CSE 2021 Set 1 | Question: 35

Question

Consider the two statements.

• $S_1:\quad$ There exist random variables $X$ and $Y$ such that $  \left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2>\textsf{Var}[X]\textsf{Var}[Y]$
• $S_2:\quad$ For all random variables $X$ and $Y, \textsf{Cov}[X,Y]=\mathbb E  \left[|X-\mathbb E[X]|\;|Y-\mathbb E[Y]|\right ]$

Which one of the following choices is correct?

• A. Both $S_1$ and $S_2$ are true
• B. $S_1$ is true, but $S_2$ is false
• C. $S_1$ is false, but $S_2$ is true
• D. Both $S_1$ and $S_2$ are false

Comments

True! 🤣🤣🤣

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This link can be very helpful in this topic : Covariance | Correlation | Variance of a sum | Correlation Coefficient: (probabilitycourse.com)

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THANKS @Abhrajyoti00

Answer 1

$S_1:$ This statement is false because correlation formulae is $r = \dfrac{\textsf{Cov}(x,y)}{\sqrt{\textsf{Var}(x)\textsf{Var}(y)} }$.
So we have $r^2\textsf{Var}(x)\textsf{Var}(y) = (\textsf{Cov}(x,y))^2.$ By comparing above we can see that the given statement is false

Ref: https://proofwiki.org/wiki/Square_of_Covariance_is_Less_Than_or_Equal_to_Product_of_Variances

$S_2:$ This statement is false since negative covariance exists (strong negative association between random variable) but RHS of the equation always gives a positive answer.

Comments

$-1 \leq \rho \leq 1 $

Correlation Coefficient ($\rho$) is always between -1 and 1.