MIT14_30s09_lec12 - MIT OpenCourseWare

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Unformatted text preview: MIT OpenCourseWare 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: . 1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 12 Konrad Menzel March 19, 2009 Properties of Medians and Percentiles We defined the median of a random variable via 1 P ( X < median( X )) = 2 When X is discrete or has point masses that generate jumps in the c.d.f., this definition may not be useful, so in the more general case, we define the median as 1 median( X ) := min m R : P ( X m ) 2 The change with respect to the narrower definition is that if the c.d.f. has a discontinuity which makes it leap over the value 1 2 , just locate the median at the point of that discontinuity. We can also define other percentiles of the distribution of X : Definition 1 For a random variable X , the quantile is given by q ( X, ) := min { q R : P ( X q } We also call q ( X, p/ 100) the p th percentile. Note that following this definition, the median corresponds to the 50th percentile. Other frequently used quantiles are deciles ( p = 10 , 20 , 30 , . . ., 90) and quartiles ( p = 25 , 50 , 75). Now we wont spend as much time on properties of quantiles as we did for expectations, but Id just like to point out two important ways in which the median behaves very differently from the expectation: For one, we saw from Jensens Inequality that for a function u ( X ), the expectation E [ u ( X )] depends a lot on the curvature of u ( x ) in the regions where the probability mass of X lies. Generally, the median median( u ( X )) will also be different from u (median( X )), but there is a notable exception: Proposition 1 Suppose u ( x ) is strictly increasing in the support of X . Then median( u ( X )) = u (median( X )) Proof: The median of X satisfies P ( X < median( X )) = 1 2 . Since u ( x ) is strictly increas- ing, the event X < m is identical to the event...
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MIT14_30s09_lec12 - MIT OpenCourseWare

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