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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
- 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.
- Let ,
and . Show that has a multivariate
normal distribution and identify the mean and variance of U.
- Use the result in class, for multivariate normals which have a density
to show that is independent of .
- Express as a function of U.
- Use the fact that if and are independent then so are
and for any functions G and H to show that
is independent of .
- Express the sample variance of the in terms of U
and use this to show that has a distribution
on 2 degrees of freedom.
- 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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Richard Lockhart
Tue Feb 11 15:10:05 PST 1997