galton - Bivariate Normal Distribution and Regression...

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Unformatted text preview: Bivariate Normal Distribution and Regression Application to Galtons Heights of Adult Children and Parents Sources: Galton, Francis (1889). Natural Inheritance, MacMillan, London. Galton, F.; J.D. Hamilton Dickson (1886). Family Likeness in Stature, Proceedings of the Royal Society of London , Vol. 40, pp.42-73. Data Heights of Adult Children and Parents Adult Children Heights are reported by inch, in a manner so that the median of the grouped values is used for each (62.2,,73.2 are reported by Galton). He adjusts female heights by a multiple of 1.08 We use 61.2 for his Below We use 74.2 for his Above Mid-Parents Heights are the average of the two parents heights (after female adjusted). Grouped values at median (64.5,,72.5 by Galton) We use 63.5 for Below We use 73.5 for Above Adult Child vs Mid-Parent Height 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 63 64 65 66 67 68 69 70 71 72 73 Mid-Parent Adult Child Mid-Parent Height 50 100 150 200 250 63.5 64.5 65.5 66.5 67.5 68.5 69.5 70.5 71.5 72.5 Height Frequency Adult Child Heights 20 40 60 80 100 120 140 160 180 61.2 62.2 63.2 64.2 65.2 66.2 67.2 68.2 69.2 70.2 71.2 72.2 73.2 74.2 Height Frequency Joint Density Function ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] 2 1 2 2 1 1 2 2 2 2 2 1 2 1 1 1 2 1 2 2 2 2 2 2 1 2 2 1 1 2 1 2 1 1 2 2 2 2 2 1 2 1 ) ( ) ( ) ( ) ( : where , 2 1 2 1 exp 1 2 1 ) , ( -- = = = = = < < - - +---- --- = Y Y E Y V Y E Y V Y E y y y y y y y y f 0.05 0.1 0.15 0.2-3-2.5-2-1.5-1-0.5 0.5 1 1.5 2 2.5 3 x1 Bivariate Normal Density 0.15-0.2 0.1-0.15 0.05-0.1 0-0.05 1 = 2 = 0 1 = 2 = 1 = 0 .4 Marginal Distribution of Y 1 (P. 1) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 [ ] 2 2 2 2 2 1 1 2 1 2 2 2 2 1 2 2 1 1 2 2 2 2 1 1 2 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 1 2 2 2 2 1 1 2 1 2 2 2 2 1 2 2 1 1 2 2 2 1 1 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 1 2 2 1 2 2 2 2 1 2 2 1 1 2 2 2 1 1 2 2 2 1 2 2 2 2 2 1 2 2 2 2 2 2 2 1 2 2 1 1 2 1 2...
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galton - Bivariate Normal Distribution and Regression...

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