Math 5p87
Lecture 12
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 13, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Theorem 13.5: approximating general rvs with simple rvs
Not clear how to derive results
Math 5p87
Lecture 16
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Review of Distributions
is the distribution of a rv X if it is a probability measure on
(R,
Math 5p87
Lecture 14
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 27, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Proof of MCT
Proving (iii) 0 Xn X implies E [Xn ] E [X ] for nite E [X ].
By (ii), limn
Math 5p87
Lecture 11
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 11, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Revisiting the independence of functions of simple rvs
(X ) = the smallest -eld that con
Math 5p87
Lecture 13
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 25, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Expectation (integration)
(, F , P) or (, F , ). X random variable or f measurable
Will
Math 5p87
Lecture 15
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 4, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
(, F , P) vs (R, R, )
rvs are dened based on an underlying triple (, F , P).
In reality, som
Math 5p87
Lecture 17
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 11, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Creating more distributions
Based on existing discrete and continuous rvs, one can
construc
Math 5p87
Lecture 21
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 25, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Betting system
The gamblers ruin is too simplistic.
Interested in:
Fn = F0 + W1 X1 + W2 X2
Math 5p87
Lecture 22
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 25, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Tree diagram and elementary conditional probabilities
2P81: Do you remember how to use a tr
Math 5p87
Lecture 20
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 20, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Gambling and gamblers ruin
iid cfw_Xi s.t. P(Xi = 1) = p and P(Xi = 1) = q = 1 p
indeed ex
Math 5p87
Lecture 18
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 13, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Proof of Theorem 20.4
Theorem 20.4. If cfw_n is a sequence of probability measure on
(R, R
Math 5p87
Lecture 19
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 18, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Veriable conditions for as
Recall that rvs Xn converge to X almost surely (a.s.) if
P[limn
Math 5p87
Lecture 7
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 28, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Proof of Theorem 4.1.(a)
Theorem 4.1(a).
P(lim inf An ) lim inf P(An ) lim sup P(An ) P(li
Math 5p87
Lecture 1
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 7, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Introduction
Perform an experiment.
Sample space - any set
An outcome = an element of
An e
Math 5p87
Lecture 6
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 28, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Independent classes (nite case)
Classes A1 , .An are independent if for Ai Ai , A1 , .An a
Math 5p87
Lecture 2
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 9, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
The easiest uncountable example
We all know what Unif[0,1] means, or do we?
Take (for conve
Math 5p87
Lecture 8
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 30, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Some more coin ipping examples
1
2
P(H i.o.) = 1 so that P(T a.a.) = 0
Let An = nth ip = (
a
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Math 5p87
Lecture 3
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 14, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Equivalence relation
A relation on S. Write x y if x is related to y .
Examples: x knows y
Math 5p87
Lecture 4
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 16, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Probability triple
F P() is a -eld if
(i) , F
(ii) If A F , then Ac F
(iii) For any counta
Math 5p87
Lecture 9
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 4, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Role of -system
P is a -system if A, B P = A B P.
A eld is a -system (e.g. B0 , F0 ). In a
Math 5p87
Lecture 10
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
More on simple rvs
(, F , P). X : R is a simple rv if it has nite range and
for each x R,
Math 5p87
Lecture 5
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 21, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p87
Last thing to check for (0, 1], B, )
Recall that we have already dened (A) on all nite uni