The inequality is a verification of jensens

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The inequality is a verification of Jensen’s inequality for this particular case. Suppose we want to predict Y using a constant. Because is the median of log Y , Med Y exp is the best predictor in the MAE sense. If we want the best MSE predictor we need to know 2 Var log Y  in addition to E log Y  . 77
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The variance of a lognormal RV can be shown to be Var Y exp 2 2  exp 2 1 E Y  2 exp 2 1 As E Y increases, so does Var Y . The lognormal distribution is used to model income distributions and durations (failure times). 78
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3 . GENERATING PARTIALLY CONTINUOUS DISTRIBUTIONS The traditional distributions are either purely discrete or purely continuous. But in economics applications, we often see partially continuous RVs. The most common kind of partially continuous RVs are nonnegative RVs – so P X 0 0 – and have probability mass at zero but are continuous for values above zero: P X 0 0 P X x 0, x 0. 79
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