Bivariate Normal Distribution and Regression

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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.

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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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Joint Density Function
m1m20 s1s21 r0.4
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Marginal Distribution of Y1 (P. 1)
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Marginal Distribution of Y1 (P. 2)
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Conditional Distribution of Y2 Given Y1y1 (P. 1)
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Conditional Distribution of Y2 Given Y1y1 (P. 2)
This is referred to as the REGRESSION of Y2 on Y1
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Summary of Results
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Heights of Adult Children and Parents

• Empirical Data Based on 924 pairs (F. Galton)
• Y2 Adult Childs Height
• Y2 N(68.1,6.39) s22.53
• Y1 Mid-Parents Height
• Y1 N(68.3,3.18) s11.78
• COV(Y1,Y2) 2.02 ? r 0.45, r2 0.20
• Y2Y1y1 is Normal with conditional mean and
  variance

y1 Unconditional 63.5 66.5 69.5 72.5
EY2y1 68.1 65.0 66.9 68.8 70.8
sY2y1 2.53 2.26 2.26 2.26 2.26
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E(Child) Parentconstant Galtons
Finding E(Child) independent of parent
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Expectations and Variances

• E(Y1) 68.3 V(Y1) 3.18
• E(Y2) 68.1 V(Y2) 6.39
• E(Y2Y1y1) 24.50.638y1
• EY1E(Y2Y1y1) EY124.50.638Y1
  24.50.638(68.3) 68.1 E(Y2)
• V(Y2Y1y1) 5.11 ? EY1V(Y2Y1y1) 5.11
• VY1E(Y2Y1y1) VY124.50.638Y1 (0.638)2
  V(Y1) (0.407)3.18 1.29
• EY1V(Y2Y1y1)VY1E(Y2Y1y1)
  5.111.296.40 V(Y2) (with round-off)