Stat 407 Lab 11 Multivariate Normal Fall 2001
In this lab we look at samples from multivariate normal distributions with different means and covariances,
and one real data set to assess normality.
1. Consider a bivariate normal population with µ1 = 0, µ2 = 2, σ11 = 2, σ22 = 1, ρ12 = 0.5.
(a) Write out the bivariate normal density function.
(b) Write out the squared generalized distance expression (X − µ)0 Σ−1 (X − µ) as a function of x1 , x2 .
2. Using the tools available in S-Plus generate and examine samples from a standard normal distribution.
Choose Data, Random Numbers, and select Normal, mean 0 and standard deviation 1.
(a) Generate samples of size 20, 100, and the number of variables equal to 2 and 5. This will give you 4
data sets.
(b) Plot the samples using pairwise plots (or a scatterplot matrix as appropriate). Describe the differences
between the 4 data sets.
(c) Plot the data using normal probability plots. Describe the differences between the 4 data sets.
3. Now start ggobi on several test data sets, and 50% contour ellipse for the distribution.
In each case note the main features of the shape.
(a) Sample from a standard 3D normal, and 50% contour from the standard normal probability density.
Look at the 1D plots of each variable in the normal sample, and describe what you see. Adjust the
smoothness parameter of the ASH curve to get the most bell-shaped curve.
Watch both data sets in the tour.
(b) Sample from a standard 6D normal, and 50% contour from the standard normal probability density.
Look at the 1D plots of each variable in the normal sample, and describe what you see. Adjust the
smoothness parameter of the ASH curve to get the most bell-shaped curve.
Watch both data sets in the tour.
4. Explore the womens track data for its marginal normality, multivariate normality, nonlinear relationships
and outliers.
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