STAT 350

Postscript version of these questions

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 $Y_1,\ldots,Y_4$. 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 Y₁,Y₂,Y₃ are a function of the first function while Y₄ is a function of the second.

(a)

Let U₁ = (2 Z₁ - Z₂ - Z₃)/3, U₂ = (2 Z₂ - Z₁ - Z₃)/3 and U₃ = (Z₁+Z₂+Z₃)/3. Show that U=(U₁,U₂,U₃)^T has a multivariate normal distribution and identify the mean and variance of U. The question I wanted to ask: Let $U_1 = (Z_1-Z_2)/\sqrt{2}$, $U_2 = (Z_1+Z_2-2Z_3)/\sqrt{6}$ and U₃ = (Z₁+Z₂+Z₃)/3. Show that U=(U₁,U₂,U₃)^T has a multivariate normal distribution and identify the mean and variance of U.

(b)

Use the result in class, for multivariate normals which have a density to show that (U₁,U₂) is independent of U₃.

(c)

Express Y₃ as a function of U.

(d)

Use the fact that if X₁ and X₂ are independent then so are G(X₁) and H(X₂) for any functions G and H to show that Y₁,Y₂,Y₃ is independent of Y₄.

(e)

Express the sample variance of the X_i, i=1,2,3 in terms of Uand use this to show that $(n-1)s_X^2/\sigma^2$ has a $\chi^2$ distribution on 2 degrees of freedom (with n=3). Note: in fact the sample variance of X₁,X₂ is a function of U₁. Generalizations of this idea can be used to develop an identity of the form

(n-1)s²_n = (n-2)s²_n-1+ U_n²

for a suitable U_n where s²_n is the sample variance for $X_1,\ldots,X_n$.

2.

In class I discussed the general formula for a multivariate normal density. Suppose that Z₁ and Z₂ are independent standard normal variables. Assume that X₁ = a Z₁ + b Z₂ + c and X₂ = d Z₁ + e Z₂ + f. Find the joint density of X₁ and X₂ by evaluating the formulas I gave in class. Express $P(X_1 \le t)$ as a double integral. I want to see the integrand and the limits of integration but you need not try to do the integral.