Multivariate distributions The Normal distribution 1 The Normal
Multivariate distributions
The Normal distribution
1. The Normal distribution – parameters m and s (or s 2) 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
The probability density of the normal distribution If a random variable, X, has a normal distribution with mean m and variance s 2 then we will write:
The multivariate Normal distribution
Let = a random vector Let = a vector of constants (the mean vector)
Let = a p p positive definite matrix
Surface Plots of the bivariate Normal distribution
Contour Plots of the bivariate Normal distribution
Scatter Plots of data from the bivariate Normal distribution
Trivariate Normal distribution - Contour map x 3 mean vector x 2 x 1
Trivariate Normal distribution x 3 x 2 x 1
Trivariate Normal distribution x 3 x 1 x 2
Trivariate Normal distribution x 3 x 2 x 1
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:
The data :
Alb, Chl, Bp
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