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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There are two common strategies to generating nonnegative RVs with mass at zero. First, suppose that W is a continuous RV taking on negative and possitive values. In fact, assume f W w 0 for all w . Define X max 0, W so X 0 if and only if W 0 and X W if W 0. It follows that P X 0 P W 0 F W 0 P X x P W x F W x , x 0. 80
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Therefore the CDF of X is F X x 0, x 0 F X x F W x , x 0 and there is a jump in F X x at zero (because P W 0 0was assumed). The PDF of X is f X x 0, x 0 f X 0 F W 0 f X x dF X dx x dF W dx x f W x , x 0. 81
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ThePDFiseasytofindgiventheCDFandPDFof W . EXAMPLE : Suppose that W ~ Normal , 2 and X max 0, W . Then F W w  w / and f W w 1  w / . Therefore, f X 0 P W 0 / 1 / f X x 1 x , x 0. 82
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The indicator function is a useful shorthand. If A is any event, the function 1 A 1if A is true and 1 A 0if A is not true. So 1 x 0 1 if and only if x 0. Then the density can be written as f X x 1 /  1 x 0 1 x 1 x 0 (where we now just ignore values of x that are not possible). This is often called the censored normal distribution (with censoring at zero). 83
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A second possibility is to specify P X 0 and then use a standard density for strictly positive random variables for x 0. A common choice is the lognormal. Of course, we must multiply this density by 1 to ensure that the total probability is one. Using indicator functions we have f X x 1 x
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The inequality is a verification of Jensens inequality for...

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