Solutions to some exercises and problems
Teck-Cheong Lim
Department of Mathematical Sciences
George Mason University
4400, University Drive
Fairfax, VA 22030
U.S.A.
e-mail address: tlim@gmu.edu
Abstract
Solutions to some exercises and problems from Stein

18.440: Lecture 18
Uniform random variables
Scott Sheeld
MIT
18.440 Lecture 18
1
Outline
Uniform random variable on [0, 1]
Uniform random variable on [, ]
Motivation and examples
18.440 Lecture 18
2
Outline
Uniform random variable on [0, 1]
Uniform random

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Failure rate
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Failure rate is the frequency with which an engineered system or component fails, expressed in
C

Department of Mathematics
Tutorial Sheet No. 6
MAL 509 (Probability Theory)
1. For arbitrary random variables, X and Y , show that
P (X + Y x + y) P (X x) + P (Y y) P (X x, Y y), x, y R
1/n
2. Show that: FX1 ,X2 ,.,Xk (x1 , x2 , . . . , xk ) [FX1 (x1 )FX2

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Ratio distribution
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A ratio distribution (or quotient distribution) is a probability distribution constructed

Informatica Economic, nr. 2 (42)/2007
132
Applying the Moment Generating Functions to the Study
of Probability Distributions
Silvia SPTARU
Academy of Economic Studies, Bucharest
In this paper, we describe a tool to aid in proving theorems about random var

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Power series
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Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A
Berthold K. P. Horn
629
Closed-form solution of absolute orientation using unit
quaternions
Berthold K. P. Horn
Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, Hawaii 96720
Receive

3.4 QR Factorization via the Householder and
Givens Transformations
The QR factorization of A is A = QR where QT Q = I and R is an upper triangular
T
matrix. Multiply A by Q
to get QT A = Q T QR = R. An alternative to the Gram-Schmidt
T
methods is to find

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Homework IV - Algebraic Topology
Due 1st March 16
Concise, coherent and rigorous solutions are encouraged. Homeworks more than one page must be stapled.
Problem 1 For (X, A) consider the long exact sequence:
k+1
qk
k1
k
k
Hk (A)
Hk (X) Hk (X, A)
Hk1 (A

Homework III - Algebraic & Differential Topology
Due 21st August 15
Concise, coherent and rigorous solutions are encouraged. Homeworks more than one page must be stapled.
NOTE - You would present problems 1 and 2 on Monday, 17th August and problems 4 and

Homework VI - Algebraic Topology
Due 5th April 16
Concise, coherent and rigorous solutions are encouraged. Homeworks more than one page must be stapled.
Problem 1 A map f : S n S n is said to be of degree k if f : Hn (S n ) Hn (S n ) is multiplication by

Homework V - Algebraic Topology
Due 21st March 16
Concise, coherent and rigorous solutions are encouraged. Homeworks more than one page must be stapled.
Problem 1 A map f : S 1 S 1 is said to be of degree n if f : H1 (S 1 ) H1 (S 1 ) is multiplication by

Homework I - Algebraic & Differential Topology
Due 28th July 15
Concise, coherent and rigorous solutions are encouraged. Homeworks more than one page must be stapled.
NOTE - You would be asked to present problems 2, 3 and 5 on Friday, 24th July.
Problem 1

Exercises 1.
1. Let L|k be a field extension and S, T L. Show that
(i) k(S)(T ) = k(S T )
(ii) k(S) is the field of fractions of k[S].
2. Prove that k(X) is not finitely generated as a k-algebra. (X is an indeterminate over k.)
3. Show that any non-zero r

Notes on Convergence of Power Series
Chris Wendl
April 12, 2004
1
Introduction
P
n
The question is this: given an infinite series of the form
n=0 an (x x0 ) , for what values of x does it
converge? This is an important thing to know, as it tells us, for

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312:;
’Proisiw! 'FS‘ {PYNLRCeJ
—‘.
E7“; 1. H9 Under the usual assumptions of the Birthday Problem, ﬁnd the expected
number of days in a year that are birthdays of exactly 19 persons in a group of n.
Exc 2. (m)- Suppose X1-,X2,.,X100 are i.i.d. random vari