Let x and y be random variables, not necessarily independent, that take real values in the interval

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GATE-CS is a competitive examination conducted in India. This question is from the 2024 GATE-CS Engineering Mathematics paper.

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Question

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 x_bar, y_bar, z_bar respectively. Which one of the following statements is TRUE?

Accepted Answer

Correct Answer: (D) z_bar Concept -- Expected Values and Bounds: Since x and y both take values in [0, 1], we have 0 Option A (z_bar = x_bar * y_bar) is only true when x and y are independent. Options B and C are not generally true: if x and y are perfectly positively correlated (x = y), then z_bar = E[x^2] >= (E[x])^2 = x_bar * y_bar (by Jensen's inequality), contradicting B. If x and y are negatively correlated, z_bar can be less than x_bar * y_bar. So neither B nor C holds universally. Only D is always true.

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 x_bar, y_bar, z_bar respectively. Which one of the following statements is TRUE?

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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 x_bar, y_bar, z_bar respectively. Which one of the following statements is TRUE?

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