galton - Bivariate Normal Distribution and Regression...

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Bivariate Normal Distribution and Regression Application to Galton’s 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.
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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”
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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
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Mid-Parent Height 0 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
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Adult Child Heights 0 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
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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 ) , ( σ σ μ μ ρ σ μ σ μ σ μ σ σ μ μ ρ σ μ ρ ρ σ σ π - - = = = = = < < - - + - - - - - - - = Y Y E Y V Y E Y V Y E y y y y y y y y f 0 0.05 0.1 0.15 0.2 -3 -2.5 -2 -1.5 -1 -0.5 0 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 μ 1 = μ 2 = 0 σ 1 = σ 2 = 1 ρ= 0.4
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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 1 1 2 2 2 2 2 1 2 2 1 1 1 1 2 1 2 1 exp 1 2 1 2 1 2 1 exp 1 2 1 : brackets square in the g subtractin and adding by exponent in the square the Completing 2 1 2 1 exp 1 2 1 : exponent in r denominato common forming and ly) (temporari constant out Bringing 2 1 2 1 exp 1 2 1 , dy y y y y y dy y y y y y y y dy y y y y
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