Lecture I17 January 2012
36217
Liz Kulka
Prof. Alessandro Rinaldo ([email protected])
OH: W, 3:00pm  4:00pm, Baker Hall 229I
Syllabus
1
Expansion on/Highlights from Syllabus
Class will be cancelled Tuesday 17 April (Carnival Week).
This course will cover
Lecture IX14 February 2012
36217
Liz Kulka
Readings: Section 2.4
Material through this lecture will be on Exam 1.
1
Discrete Random Variables
X Bernoulli(p)
X Binomial(n, p)
0<p<1
X Geometric(p)
X N egBin(r, p)
X HyperGeom(R, B, n) or (N, R, n) N = B + R
Lecture VIII9 February 2012
36217
Liz Kulka
Readings:
1
Random Variables, Continued
1.1
Review
Random Variables X : R
Range(X) is the set of all possible values of X
pmf (probability mass function) of X, pX : pX (x) = Pr(cfw_X = x), x RAN (X)
Bernoul
Lecture VII7 February 2012
36217
Liz Kulka
Readings:
Midterm 1 in 10 days, no notes and no calculators permitted.
There will be a handout on BB with equations that we do not need to memorize.
The format will be identical to the practice exam and the dicu
Lecture VI2 February 2012
36217
Liz Kulka
Readings:
1
Recall
Pr(AB) =
Pr(A B)
Pr(B)
(Will be on the exam!)
P a probability on and A1 , A2 , . . . , An a partition of with P (Ai ) > 0) i.
Law of Total Probability:
n
P (B) =
P (BAi )P (Ai )
i=1
Bayes The
Lecture V31 January 2012
36217
Liz Kulka
Readings: 1.3  1.5
1
Conditional Probability
Recall the scenario where we took some medicine that had possible side eects.
A = cfw_I will experience side eects
B = cfw_I am a smoker
Pr(A) = 0.1 Unconditional Prob
Lecture IV26 January 2012
36217
Liz Kulka
Readings: 1.6
1
Counting Methods
Discrete Uniform Distribution: If is nite and all points in are equally likely, then
Pr(A) =
A
A

This is a type of Random Sample.
Four counting methods:
1.1
Permutations
Cons
Lecture III24 January 2012
36217
Liz Kulka
Readings: 1.1, 1.2, 1.6
1
Review
Given a sample space , a probability law or distribution P is a function dened over subsets of
such that:
1. P(A) 0 A
2. P() = 1
3. If A1 , A2 , . . . is a sequence of disjoint
Lab 1 TV Jammer
* Please read and understand the Lab Readiness Prerequisites and
Lab Etiquette and Procedures documents before starting this lab *
1. Introduction
In this lab you will build the core pieces of a TV jammer. By jammer, we mean a nearunivers
10/20/2014
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advertisement
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peers/friends/family
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21257 Test 1 Notes
1.
Chapter 1.4 Solving LP Graphically
a. Make constraints equalities, and graph the corresponding lines
b. Shade in graph to match inequality constraints
c. Find the corners
d. Evaluate the corners for the solution. Solution must be a
214 Exam 1
1) Lecture 1: Introduction
A. Quality attributes describe the codes fitness for further development/use:
reuse code, easy to change, easy to add
B. Objects support simulation, extensibility, and modifiability
C. Method calls are dispatched to m
Lecture XXV1 May 2012
1
36217
Liz Kulka
Final Exam
Monday, May 7, 1:004:00pm DH2210
Same format as Midterms
Cumulative Exam
Practice Exams on Blackboard
Chapter 5: Probability Inequalities And Limit Theorems
2
Probability Inequalities
Want to comput
Lecture XXVI3 May 2012
1
36217
Liz Kulka
Limit Theorems: Convergence in Probability
Denition: A sequence X1 , X2 , . . . of random variables converges in probability to c R, written
as Xn p c, if for all > 0 and > 0, n( , ) such that Pr(Xn c > ) < for
Lecture II19 January 2012
36217
Liz Kulka
Readings: Sections 1.1 and 1.2
Oce Hours Wednesdays 23PM
HW 1 posted tonight
1
Denition of Probability and its Properties
1.1
Set Theory
Set theory is necessary to dene probability properly. Set theory is the fu
Lecture XXIV26 April 2012
1
36217
Liz Kulka
Stationary Distribution
Limiting Distribution: limn ri,j (n). If j is a transient state the limit will be 0. Sometimes
the limit is not dened.
For irreducible, aperiodic markov chains, this is the stationary di
Lecture XXIII24 April 2012
1
36217
Liz Kulka
Discrete Time, Finite State Space, Time Homogeneous First
Order Markov Chains
Sequence of rvs: X0 , X1 , X2 , . . . s.t. range(Xn ) = S = cfw_1, . . . , m n.
First order Markov Chain:
Pr(Xn = in Xn1 = in1 ,
Lecture XXII12 April 2012
1
36217
Liz Kulka
Discrete Time, Finite State Space Order One Homogeneous
Markov Chains
Sequence of rvs: X0 , X1 , X2 , . . .
Pr(Xn = in Xn1 = in1 , . . . , X0 = i0 ) = Pr(Xn = in Xn1 = in=1 ) = pi,j
independent of n. Also ass
Lecture XX5 April 2012
1
36217
Liz Kulka
Covariance
Let X and Y be two rvs. The covariance of X and Y is
Cov [X, Y ] = E [(X E [X])(Y E [Y ])]
If Cov [X, Y ] > 0 X and Y tend to move together:
y
x
If Cov [X, Y ] < 0 X and Y tend to move together:
y
x
E
Lecture XXI10 April 2012
1
36217
Liz Kulka
Markov Chains
A stochastic process in discrete time is a sequence X0 , X1 , . . . of random variables.
1.1
Trivial Cases
X0 , X1 , X2 are iid from a common distribution if X0 Bernoulli(p) (a Bernoulli process).
Lecture XIX3 April 2012
36217
Liz Kulka
Readings:
Score
[90, 130)
[80, 90)
Midterm 2:
[70, 80)
[60, 70)
[0, 60)
1
Percentage of class
50%
16%
12%
11%
11%
Conditioning
X, Y continuous r.v.s the conditional pdf of X given Y = y is:
fXY (y) : R R
such tha
Lecture XVIII27 March 2012
1
36217
Liz Kulka
Joint, Marginal and Conditional pdf s
X, Y are uniformly distributed over set A R2
4
A
3
2
L
1
1
2
3
4
1
4
fX,Y (x, y) =
0
(x, y) A
o.w.
Pr(X, Y ) S) =
fX,Y (x, y)dxdy
(x,y):(x,y)S
Pr(X = Y ) =
fX,Y (x, y)dxd
Lecture XVII22 March 2012
1
36217
Liz Kulka
Review of Integration
Let a b +:
b
xc dx =
a
b
1
xc+1
c+1
c
f (x)dx =
a
b
(1)
a
b
f (x)dx +
a
f (x)dx if a < c < b
(2)
c
b
b
cf (x) + dg(x)dx =
a
b
cf (x)dx +
a
dg(x)dx
(3)
a
Integration by Parts:
b
b
f (x)g (x