GATE 2020 ME (ME1) – Question 45
X and Y are exponentially distributed random variables, each of mean 0.50. Let Z = X + Y and r be the correlation coefficient of X and Y. If Var(Z) = 0, find r to two decimals.
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Correct answer: -1 to -.98
Explanation
Each exponential variance is mean²=0.25. Formally $0=0.25+0.25+2r\sqrt{0.25(0.25)}=0.5+0.5r$, giving **r=−1.00**. However, exact exponential marginals cannot sum to a constant because they are unbounded and nonnegative. This is the intended variance-identity answer, with an inconsistent joint-distribution premise.