Multivariate distributions

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Presentation transcript:

Multivariate distributions

The Normal distribution

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

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:

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 x3 mean vector x2 x1

Trivariate Normal distribution x3 x2 x1

Trivariate Normal distribution x3 x2 x1

Trivariate Normal distribution x3 x2 x1

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