1. Probability Distributions
A Brief Introduction

2. Normal (Gaussian) Distribution
Bell-shaped distribution with tendency for individuals to clump around the group median/mean
Used to model many biological phenomena
Many estimators have approximate normal sampling distributions (see Central Limit Theorem)
Notation: Y~N(m,s2) where m is mean and s2 is variance
Obtaining Probabilities and Quantiles in R:
To obtain: F(y)=P(Y≤y) Use Function: pnorm(y,m,s)
To obtain the pth quantile: P(Y≤yp)=p Use Function: qnorm(p, m,s)
Virtually all statistics textbooks give the cdf (or upper tail probabilities) for standardized normal random variables: z=(y-m)/s ~ N(0,1)

3. Normal Distribution – Density Functions (pdf)

4. Second Decimal Place of z
Integer part and first decimal place of z

5. Chi-Square Distribution
Indexed by “degrees of freedom (n)” X~cn2
Z~N(0,1)  Z2 ~c12
Assuming Independence:
Obtaining Probabilities in R:
To obtain: 1-F(x)=P(X≥x) Use Function: pchisq(x,n)
To obtain quantiles: P(X≤xp)=p Use Function: qchisq(x,n)
Virtually all statistics textbooks give upper tail cut-off values for commonly used upper (and sometimes lower) tail probabilities

6. Chi-Square Distributions

7. Critical Values for Chi-Square Distributions (Mean=n, Variance=2n)

8. Student’s t-Distribution
Indexed by “degrees of freedom (n)” X~tn
Z~N(0,1), X~cn2
Assuming Independence of Z and X:
Obtaining Probabilities /Quantiles in EXCEL:
To obtain: F(t)=P(T≤t) pt(t,n) To obtain: pth quantile: qt(p,n)
Virtually all statistics textbooks give upper tail cut-off values for commonly used upper tail probabilities

10. Critical Values for Student’s t-Distributions E(T)=0 (n>1) V(T)=n/(n-2)

11. F-Distribution Indexed by 2 “degrees of freedom (n1,n2)” W~Fn1,n2
X1 ~cn12, X2 ~cn22
Assuming Independence of X1 and X2:
Obtaining Probabilities/Quantiles in R:
To obtain: F(w)=P(W≤w): pf(w,n1,n2) pth quantile: qf(p,n1,n2)
Virtually all statistics textbooks give upper tail cut-off values for commonly used upper tail probabilities

13. Critical Values for F-distributions P(F ≤ Table Value) = 0.95

14. Multivariate Normal Distribution

15. Results Involving Multivariate Normal - I

16. Results Involving Multivariate Normal - II

17. Results Involving Multivariate Normal - III

18. Multivariate Normal Likelihood Function

19. Maximum Likelihood Estimator of m

20. Maximum Likelihood Estimator of S

21. Results for ML Estimators and Large-Sample Properties

22. 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”

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

27. Marginal Distribution of Y (p. 1)

28. Marginal Distribution of Y (p. 2)

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

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

31. Summary of Results

32. 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) =  r = 0.45, r2 = 0.20
Y2|Y1=y1 is Normal with conditional mean and variance: y1
Unconditional
63.5
66.5
69.5
72.5
E[Y2|y1]
68.1
65.0
66.9
68.8
70.8
sY2|y1
2.53
2.26

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

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