tut6_10

tut6_10 - . 00 . 00-. 25-. 25-. 00-. 00 . 00 . 00 . 25 ....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
The University of Western Ontario Department of Statistical and Actuarial Sciences Statistical Sciences 3859a Tutorial 6 1. Suppose ε 1 and ε 2 are independent standard normal random variables. (a) Identify the distribution of Y = 5 ε 1 +2 ε 2 , specifying any relevant parameter values. (b) Identify the distribution of Z = ε 2 1 + ε 2 2 , specifying any relevant parameter values. 2. Suppose X is a 30 × 6 matrix consisting of linearly independent columns. (a) Show that A = X ( X T X ) - 1 X T is a symmetric, idempotent matrix. (b) Find the trace of A . (c) Suppose ε is a vector consisting of 30 independent standard normal random vari- ables. Show that if B = ε T and C = ε T ( I - A ) ε , then B and C are independent. (d) Identify the distributions of B , C and D = 4 B/C , specifying any relevant parameter values. 3. Let ε denote a vector of eight independent standard normal random variables, and let A = 0 . 25 0 . 25 0 . 00 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 25 0 . 25 0 . 00 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 00 0 . 00 0 . 25 0 . 25 - 0 . 25 - 0 . 25 - 0 . 00 - 0 . 00 0 . 00 0 . 00 0 . 25 0 . 25 - 0 . 25 - 0 . 25 - 0 . 00 - 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 25 0 . 25 0 . 00 0 . 00 - 0 . 00 - 0 . 00 - 0 . 25 - 0 . 25 0 . 25 0 . 25
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . 00 . 00-. 25-. 25-. 00-. 00 . 00 . 00 . 25 . 25-. 25-. 25-. 00-. 00 . 00 . 00 . 25 . 25 , and B = 1 8 aa T where a T = [1 1 1 1-1-1-1-1]. (a) Is A symmetric? (b) Compute A 2 . (c) Identify the respective distributions of Y 1 = T A , and Y 2 = T ( I-A ) . (d) Are Y 1 and Y 2 independent? Hence, or otherwise, identify the distribution of 3 Y 1 /Y 2 . (e) Compute E [ Y 1 ] and E [ Y 2 ]. (f) Is B idempotent? (g) Compute AB and determine ( A-B ) 2 . (h) Identify the respective distributions of W 1 = T B and W 2 = T ( A-B ) . (i) Are W 1 and W 2 independent? Hence, or otherwise, identify the distribution of W 1 /W 2 ....
View Full Document

Ask a homework question - tutors are online