List of practice Questions

Let \(X\) be a random variable which takes values in the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\).
Further, \(\Pr(X=1) = \Pr(X=2) = \Pr(X=5) = \Pr(X=7) = \frac{1}{6}\) and \(\Pr(X=3) = \Pr(X=4) = \Pr(X=6) = \Pr(X=8) = \frac{1}{12}\).
The expected value of \(X\), denoted by \(E[X]\), is equal to ___________. (rounded off to two decimal places)

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