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Bivariate Normal Distribution and Regression

This study explores the application of bivariate normal distribution and regression analysis using the heights of adult children and their parents, as originally reported by Francis Galton. It delves into empirical data involving 924 pairs, offering insights into the relationship between parent and child heights. The findings highlight key statistical parameters such as means, variances, and the regression of child height on parent height, demonstrating how parental height influences offspring stature while maintaining the independence of certain expectations and variances.

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Bivariate Normal Distribution and Regression

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  1. 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.

  2. 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”

  3. Joint Density Function m1=m2=0 s1=s2=1 r=0.4

  4. Marginal Distribution of Y1 (P. 1)

  5. Marginal Distribution of Y1 (P. 2)

  6. Conditional Distribution of Y2 Given Y1=y1 (P. 1)

  7. Conditional Distribution of Y2 Given Y1=y1 (P. 2) This is referred to as the REGRESSION of Y2 on Y1

  8. Summary of Results

  9. Heights of Adult Children and Parents • Empirical Data Based on 924 pairs (F. Galton) • Y2 = Adult Child’s Height • Y2 ~ N(68.1,6.39) s2=2.53 • Y1 = Mid-Parent’s Height • Y1 ~ N(68.3,3.18) s1=1.78 • COV(Y1,Y2) = 2.02  r = 0.45, r2 = 0.20 • Y2|Y1=y1is Normal with conditional mean and variance:

  10. E(Child)= Parent+constant Galton’s Finding E(Child) independent of parent

  11. Expectations and Variances • E(Y1) = 68.3 V(Y1) = 3.18 • E(Y2) = 68.1 V(Y2) = 6.39 • E(Y2|Y1=y1) = 24.5+0.638y1 • EY1[E(Y2|Y1=y1)] = EY1[24.5+0.638Y1] = 24.5+0.638(68.3) = 68.1 = E(Y2) • V(Y2|Y1=y1) = 5.11  EY1[V(Y2|Y1=y1)] = 5.11 • VY1[E(Y2|Y1=y1)] = VY1[24.5+0.638Y1] = (0.638)2 V(Y1) = (0.407)3.18 = 1.29 • EY1[V(Y2|Y1=y1)]+VY1[E(Y2|Y1=y1)] = 5.11+1.29=6.40 = V(Y2) (with round-off)

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