STAT 350: 95-3

Assignment 4

Part A

From text page 254-255, 6.15 c, d, e,,f, g, 6.16, 6.17. Page 257 6.25 and 6.26. Page 324 7.46. Page 394 9.11. Page 398 9.25.

Part B

1. In assignment 2 in question 2 you dealt with variables . In class I stated that the if the covariance between two components of a multivariate normal vector is 0 then the components are independent, but I indicated a proof only when the multivariate normal distribution in question has a density. In this case the variance matrix is singular so there is no density. However, in terms of the original Z it is possible to find two indendent functions of Z such that are a function of the first function while is a function of the second.
   1. Let , and . Show that has a multivariate normal distribution and identify the mean and variance of U.
   2. Use the result in class, for multivariate normals which have a density to show that is independent of .
   3. Express as a function of U.
   4. Use the fact that if and are independent then so are and for any functions G and H to show that is independent of .
   5. Express the sample variance of the in terms of U and use this to show that has a distribution on 2 degrees of freedom.

2. In class I discussed the general formula for a multivariate normal density. Suppose that and are independent standard normal variables. Assume that and . Find the joint density of and by evaluating the formulas I gave in class. Express as a double integral. I want to see the integrand and the limits of integration but you need not try to do the integral.

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