The GATE Grind

GATE 2021 CS – Question 45

Engineering Mathematics · Probability and Statistics · 2 marks · Multiple choice

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

  1. Both $S_1$ and $S_2$ are true.
  2. $S_1$ is true, but $S_2$ is false.
  3. $S_1$ is false, but $S_2$ is true.
  4. Both $S_1$ and $S_2$ are false.

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Show answer and explanation

Correct answer: (D) Both $S_1$ and $S_2$ are false.

Explanation

Cauchy-Schwarz gives Cov^2 <= Var[X]Var[Y], so S1 is false. Covariance can be negative but the expectation of absolute values is non-negative, so S2 is false.