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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 Lets go back to the data on extra-marital affairs, and look at the variables we are actually most interested in: the number of affairs during the last year, Z , and self-reported 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 extra-marital 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 extra-marital 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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