George Liu 41788974  2011
Sigma algebra and filtration = : . . ,2,1,0 =
a collection of subsets of , is a sigmaalgebra on if:
1. , 2. , 3.
Contidional Expectation: () =  1
Consider all  where event
= + = =
sub
is a
martingale if ( ) < a
NCD Systems
To adjust the transition matrix: Pr(Claim) = Pr(Claim  Loss) Pr(Loss).
Expected Claims = Pr(Claim) (Avg. claim size  Loss > x) where x is the
minimum loss size needed to make a claim (calculate from table).
NCD category at time 0
Year 1
Prem
Stochastic Processes: Let , , be a probability space with , drift coefficient, , diffusion coefficient
information structure given by = , 0,
Geometric Brownian Motion: = + OR
= exp = +
Information structure defined by a sequence of partitions:
At time
I
Brownian motion aka Wiener Process: SDE

cfw_I . I cfw_ cfw_ cfw_ . Jcfw_ $ cfw_ . Jcfw_ H cfw_ cfw_
I"
I cfw_
Increments are independent and indept of I i.e Jcfw_I , I . I cfw_
$ cfw_
Standard B.M:
"
Jcfw_ cfw_
and
(i.e Set
H cfw_ cfw_
cfw_J cf
Discrete Mathematics  Math 210  Schedule
Date
Wed.
Topic
9/2/2015 Intro; Talking, writing math
Reading/work to be
done before class
Comments
None
Fri.
Mon.
Wed.
Fri.
9/4/2015 Math sentences & sets
9/7/2015 Labor Day
9/9/2015 Relations and Functions
9/11
First Discrete Math Exam  Due Monday 9/28/15 at 8 a.m.
You may return this exam to me in person or by email.
Please do NOT send me a PDF.
This is an open book exam. The following and only the following are permissable sources.
Your text book
Your notes
A
Typing Math on a computer
Here are some common symbols. It is fine to copy them to the top of a page and copy each symbol as
you need it. Of course, you can also use LaTex or MathType.
Often, the easiest way to get a symbol is from the equation editor ava
Problems to practice on with quantified logic
Symbols I use a lot (here for convenience)
~
1. Using the college example:
Let the domain of definition be all students (x, y, z.) and courses (c, d, e.) at Simmons
College.
F(x) = x is a freshman
S(x) = x i
Second Discrete Math Exam  Fall 2015 Due on Wed. 10/21/15 at 8 a.m.
Name_
This is a take home exam. You may use your text book, the notes from class, and the Voice
Threads, but your should reference them (e.g. by Theorem 1.2.3. p45 or by VT for section 1
Theorem from before: If a, b >0 and ab then a b
x
and x
Definition
Prop. 0
Proof:
x is the largest integer less than or equal to x
x is the samlles integer greater than or equal to x.
x
x
x
Prop. 1 If x is an integer then x =
x
= x
Proof:
Prop. 2
Problems for Section 4.1  Review Problems from 3.3
Write the following using quantifiers: (You should be able to do this in under 1 minute each!)
1. If an integer x is divisble by 2 and by 3 then it is divisible by 6.
2. There is a unique number e in the
Problems for Face to Face Session for Section 5.2 statements to be proved by induction
Practice problem #10: For any real numbers x and r and any n >0:
xn + xn1r + . +xrn1 + rn = [xn+1 rn+1] / [x r]
Practice problem #14 Let p1, p2,.pn , n 2 be statement
Problems for Face to Face Session for Section 4.1
Prove or give a counterexample:
a. If a, b are real numbers and a < b then a2 < b2
b. If m and n are odd integers then m + n is even.
c. If m and n are odd integers then their average is even.
d. If m + n
MATH 115 Number Systems & Algebra for Elementary School Teachers
SYLLABUS for FALL 2013
Course meeting times/place:
TuTh 8  9:20 a.m., Room: C218 (Main College Building)
Instructor: Donna Beers
Office: S212
Email: donna.beers@simmons.edu
Phone: 6175212