Marginal and conditional distributions
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Define a conditional median of Y given X = x to be any median of the conditional distribution of Y given X = x. Suppose that we will get to observe X and then we will need to predict Y. Suppose that we wish to choose our prediction d(X) so as to minimize mean absolute error (MAE), E(|Y - d(X)|). Prove that d(x) should be chosen to be a conditional median of Y given X = x. You can use the fact that for a univariate random variable MAE is minimized at the median, m.
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