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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 8 Konrad Menzel March 3, 2009 Conditional p.d.f.s Definition 1 The conditional p.d.f. of Y given X is f XY ( x,y ) f Y  X ( y  x ) = f X ( x ) Note that if X and Y are discrete, P ( Y = y  X = x ) f Y  X ( y  x ) = P ( X = x ) which just corresponds to the conditional probability of the event corresponding to X = x given Y = y as defined two weeks ago. Note that • for a particular value of the conditioning variable, the conditional p.d.f. has all the properties of a usual p.d.f. (i.e. positive, integrates to 1) • the definition generalizes to any number of random variables on either side Example 1 Let’s go back to the data on extramarital affairs, and look at the variables we are actually most interested in: the number of affairs during the last year, Z , and selfreported ”quality” of the marriage, X . The joint p.d.f. is given by Since three quarters of respondents reported not having had Z f XZ 1 2 f X 1 17 . 80% 4 . 49% 6 . 82% 29 . 12% X 2 24 . 29% 3 . 83% 4 . 16% 32 . 28% 3 32 . 95% 3 . 33% 2 . 33% 38 . 60% f Z 75 . 04% 11 . 65% 13 . 31% 100 . 00% an affair, it might be more instructive to look at the p.d.f. of the number of affairs Z conditional on the rating of marriage quality. Conditional on the low rating, X = 1 , we have f XZ (1 , 0) 17 . 80% f Z  X (0  1) = = = 61 . 13% f X (1) 29 . 12% 1 2 f Z  X 0 Z 1 2 1 X 2 3 61 . 13% 15 . 42% 23 . 42% 75 . 25% 11 . 86% 12 . 88% 85 . 36% 8 . 63% 6 . 04% Putting the conditional c.d.f.s for the values of X = 1 , 2 , 3 together in a table, we get Why is this exercise interesting?  while in the table with the joint p.d.f., the overall picture was not very clear, we can see that for lower values of marriage quality X , the conditional p.d.f. puts higher probability mass on higher numbers of affairs. Does this mean that dissatisfaction with marriage causes extramarital affairs? Certainly not: we could just do the reverse exercise, and look at the conditional p.d.f. of reported satisfaction with marriage, X , given the number of affairs, Z . E.g. f XZ (1 , 0) 17 . 80% f X  Z (1 , 0) = = = 23 . 72% f Z (0) 75 . 04% or, summarizing the conditional p.d.f.s in a table: We see that the conditional p.d.f. of X given a larger number of affairs, Z , puts more probability on lower Z f X  Z 1 2 1 23 . 72% 38 . 54% 51 . 24% X 2 32 . 37% 32 . 88% 31 . 25% 3 43 . 91% 28 . 58% 17 . 51% satisfaction with the marriage. So we could as well read the numbers as extramarital affairs ruining the relationship. This is often referred to as ”reverse causality”: even though we may believe that A causes B, B may at the same time cause A....
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 Fall '09
 Economics, Probability, Probability theory, conditional p.d.f.

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