Discussion
Let \(x\) and \(y\) be random variables, not necessarily independent, that take real values in the interval [0,1]. Let \(z = xy\) and let the mean values of \(x, y, z\) be \(\bar{x}, \bar{y}, \bar{z}\), respectively. Which one of the following statements is TRUE?
Answer: D
The true statement is \(\bar{z} \leq \bar{x}\).
Because \(0 \leq y \leq 1\) and \(x \geq 0\), every outcome has \(z = xy \leq x\). Taking means preserves the inequality, so \(\bar{z} \leq \bar{x}\) always.
The other relations depend on the dependence between x and y, since \(\bar{z} - \bar{x}\bar{y}\) is the covariance. If \(y = x\), \(\bar{z}\) is the mean of \(x^{2}\), which is at least \(\bar{x}^{2}\), so '\(\leq\)' fails. If \(y = 1 - x\), the covariance is negative, so '\(\geq\)' fails, and equality holds only for uncorrelated variables.
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