2. Multivariate distributions

3. The Normal distribution

4. 1.The Normal distribution – parameters m and s (or s2)
Comment: If m = 0 and s = 1 the distribution is called the standard normal distribution
Normal distribution with m = 50 and s =15
Normal distribution with m = 70 and s =20

5. The probability density of the normal distribution
If a random variable, X, has a normal distribution with mean m and variance s2 then we will write:

6. The multivariate Normal distribution

7. Let
= a random vector
Let
= a vector of constants (the mean vector)

8. Let
= a p  p positive definite matrix

9. Surface Plots of the bivariate Normal distribution

10. Contour Plots of the bivariate Normal distribution

11. Scatter Plots of data from the bivariate Normal distribution

12. Trivariate Normal distribution - Contour mapx3
mean vector x2 x1

13. Trivariate Normal distributionx3 x2 x1

14. Trivariate Normal distributionx3 x2 x1

15. Trivariate Normal distributionx3 x2 x1

16. example
In the following study data was collected for a sample of n = 183 females on the variables
Age,
Height (Ht),
Weight (Wt),
Birth control pill use (Bpl - 1=no pill, 2=pill)
and the following Blood Chemistry measurements
Cholesterol (Chl),
Albumin (Abl),
Calcium (Ca) and
Uric Acid (UA). The data are tabulated next page:

17. The data :

19. Alb, Chl, Bp