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 \(\overline{x}, \overline{y}, \overline{z}\), respectively. Which one of the following statements is TRUE?
Answer
(D)
Explanation
Because \(0 \le y \le 1\) and \(x \ge 0\), every outcome has \(z = xy \le x\). Taking expectations keeps the inequality, so \(\overline{z} \le \overline{x}\) always holds.
Options (A) to (C) compare \(E[xy]\) with \(E[x]\,E[y]\); the difference is the covariance of \(x\) and \(y\), which can have either sign when they are dependent. Take \(x\) uniform on \([0,1]\):
- \(y = x\): \(\overline{z} = E[x^{2}] = \tfrac{1}{3} > \tfrac{1}{4} = \overline{x}\,\overline{y}\), so (A) and (B) fail.
- \(y = 1 - x\): \(\overline{z} = \tfrac{1}{2} - \tfrac{1}{3} = \tfrac{1}{6} < \tfrac{1}{4} = \overline{x}\,\overline{y}\), so (C) fails.
Answer: D