MAT 540 - Spring 2012
Homework 3 Solutions
Ryan Szypowski
Due May 8, 2012
1. Suppose we take two independant measurements, y1 and y2 , of a single RV x with
variance 2, where the measurement variances are 1 and 4. What is the weighted leastsquares estimat
MAT 540 - Spring 2012
Homework 1 Solutions
Ryan Szypowski
Due April 17, 2012
1. Let
A=
a b
b c
be a 2 2 real, symmetric matrix. Show that all eigenvalues of A are real. What values
of b guarantee that it is positive denite.
Solution: The eigenvalues of A
MAT 540 - Spring 2012
Homework 3
Ryan Szypowski
Due May 8, 2012
1. Suppose we take two independant measurements, y1 and y2 , of a single RV x with
variance 2, where the measurement variances are 1 and 4. What is the weighted leastsquares estimate of x in
MAT 540 - Spring 2012
Homework 2 Solutions
Ryan Szypowski
Due April 26, 2012
1. Prove that (, F, P ) dened below is a proper probability space:
= [0, 1),
F = cfw_, [0, 0.5), [0.5, 1), [0, 1),
P ([a, b) = b2 a2 .
Solution: In order to show that (, F, P )
MAT 540 - Spring 2012
Homework 2
Ryan Szypowski
Due April 26, 2012
1. Prove that (, F, P ) dened below is a proper probability space:
= [0, 1),
F = cfw_, [0, 0.5), [0.5, 1), [0, 1),
P ([a, b) = b2 a2 .
2. Prove the following statements (for any probabili
MAT 540 - Spring 2012
Homework 1
Ryan Szypowski
Due April 17, 2012
1. Let
A=
a b
b c
be a 2 2 real, symmetric matrix. Show that all eigenvalues of A are real. What values
of b guarantee that it is positive denite.
2. Let A and B be n n. Compute
Tr(A)
A
a