The mean is finite only if 1 and then it is given by

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The mean is finite only if 1, and then it is given by E X 1 40
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The second moment is finite only if 2, in which case the variance is Var X  2 2  1 2 41
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2 . 2 . Distributions on the Unit Interval Suppose that X 0,1 with P X x for all x , which means we can define the support of X as or . (That is, we can define a density to be nonzero over or .) We already studied the Uniform distribution. This distribution is often unrealistic because it gives the same probability to equal-length intervals, regardless of where they are located in . 42
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The beta distribution allows a variety of shapes. For parameters , 0, its density is usually written as f x Γ Γ Γ x 1 1 x 1 ,0 x 1 We write X ~ Beta , . 43
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The expectation can be obtained by writing the integral partly in terms of Beta 1, : E X 0 1 x Γ Γ Γ x 1 1 x 1 dx 0 1 Γ Γ Γ x 1 x 1 dx Γ Γ Γ 1 Γ 1 0 1 Γ 1 Γ 1 Γ x 1 x 1 dx Γ Γ Γ 1 Γ 1 1 44
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We have used Γ r 1 r Γ r for r and r . A similar trick can be used to find E X 2 . After tedious algebra we can also get the variance: Var X  1  2 When 2 the density is f x 6 x 1 x for 0 x 1. 45
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0 .5 1 1.5 f(x) 0 .2 .4 .6 .8 1 x PDF of the Beta(2,2) Distribution . rangex01 1000 obs was 0, now 1000 . gen fx 6*x*(1-x) . twoway (line fx x) For other values the PDF is a U-shape, such as 1/2. It can 46
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also have an everywhere positive slope and convex shape ( 1, 1) or negative slope with convex shape ( 1, 1). Concavity is possible, too 1, 1or 1, 1). 47
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0 1 2 3 f(x) 0 .2 .4 .6 .8 1 x PDF of the Beta(2,1) Distribution . rangex01 1000 obs was 0, now 1000 . gen fx 3*(x^2) . twoway (line fx x) 48
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2 . 3 . The Normal Distribution The normal distribution is the most widely used for modeling random variables that can take a wide range of values, including (in principle) negative values. It is especially useful in economic models for modeling the distribution of unobserved factors that affect behavior (such as native intelligence). Much of the exact statistical theory of multiple regression hinges on normality.
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The mean is finite only if 1 and then it is given by E X 1...

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