Problems in Probability

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Fall 2011 Strategic Practice 5: Section 2 (Seeking Sublime Symmetry) - Question 2
Explain why P(X < Y) = P(Y < X) if X and Y are i.i.d. Does it follow that P(X < Y) = 1/2? Is it still always true that P(X < Y) = P(Y < X) if X and Y have the same distribution but are not independent?
Solution: If X and Y are i.i.d., then P(X < Y) = P(Y < X) by symmetry: we can interchange X and Y since both are the probability of one draw from the distribution being less than another, independent draw. In the discrete case, P(X < Y) < 1/2 since P(X = Y) > 0. In the continuous case, P(X < Y) = 1/2 since P(X = Y) = 0. If X and Y are not independent, then it is not necessarily true that P(X < Y) = P(Y < X) since they may be structured in a way that tends to make X less than Y .
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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