Statistics 4M03/6M03 - Assignment #2

2001-10-20

Due: 2001-11-12 17:00

You should use Splus or R to do the calculations and draw the graphs.

The numbered Exercises are from Johnson & Wichern, Applied Multivariate Statistical Analysis, 4th edition.

Part A

Question 1

(a) Draw a perspective plot of the bivariate normal

probability density function.

(b) Find and graph the marginal density of X1 and the conditional density of X1 given X2 = x2, for x2 = 8, 10, 12.

(c) Plot the ellipses of concentration for 50%, 90%, 95% and 99% concentration. Add both regression lines to this plot.

Question 2

Repeat Question 1 for a bivariate distribution uniform on an ellipse, with the same mean and covariance matrix as the bivariate normal distribution in Question 1.

[Hint: Find the mean and covariance matrix for a random variable V with distribution uniform on the unit circle centred at the origin, then find the distribution of X = m + BV, then find what B you need to give X the specified covariance matrix.]

Question 3

Consider the bivariate density function

f(x1, x2) = x1 (x1 - x2)/8 when 0 < x1 < 2, -x1 < x2 <x1 and 0 otherwise.

(a) Find and graph the marginal and conditional distributions.

(b) Find and graph the regression of X1 on X2 and the regression of X2 on X1.

Part B

4.4

4.10 - 4.14

4.35 Also, see if Box-Cox transformations will help to improve normality.

Statistics 4M03