Let \(x\) and \(y\) be random variables, not necessarily independent, that…
2024
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?
- A.
\( \overline {z} = \overline {x} \overline {y}\) - B.
\( \overline {z} \leq \overline {x} \overline {y}\) - C.
\( \overline {z} \geq \overline {x} \overline {y}\) - D.
\( \overline {z} \leq \overline {x}\)
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Correct answer: D
Answer: The true statement is E[xy] ≤ E[x].
Reason: For every outcome, 0 ≤ y ≤ 1, so xy ≤ x. Taking expectations of both sides gives E[xy] ≤ E[x]. By the same argument E[xy] ≤ E[y].
Why the equality E[xy] = E[x]E[y] is not universally true: This holds if x and y are independent. For example, if x = y and P(x=1)=P(x=0)=1/2, then E[xy] = 1/2 but E[x]E[y] = 1/4.
Why E[xy] ≤ E[x]E[y] is not always true: With positive correlation E[xy] can exceed E[x]E[y]; the same x = y example gives E[xy] = 1/2 > 1/4 = E[x]E[y].
Why E[xy] ≥ E[x]E[y] is not always true: With negative correlation E[xy] can be smaller. For example, let x be 1 with probability 1/2 and 0 otherwise, and set y = 1 - x. Then E[xy] = 0 < 1/4 = E[x]E[y].
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