11/8/10
1
Probability Basics
Lecture 19
What we’ll cover
•
Tying up loose ends
•
Bayes’ Theorem
•
Random Variables
•
Expectation
•
Bounds: Bonferonni, Markov, Chebyshev’s
Inequalities
Inclusion Exclusion Principle
•
The
inclusion–exclusion principle
states that if
A
and
B
are two (finite) sets, then
•
Similarly, for three sets
A
,
B
and
C
,
•
For the general case of the principle, let
A
1
, ...,
A
n
be finite sets. Then
Example: Modeling Energy Saving
Transmissions in Wireless Networks
One node transmits on a random k out of N
timeslots, the second node listens on a
random k out of N timeslots. They are idle in
the remaining N‐k timeslots. What is the
probability that 2
nd
node hears the first?
Michael J. McGlynn, Steven A. Borbash.
Birthday Protocols
for Low Energy Deployment and Flexible Neighbor
Discovery in Ad Hoc Wireless Networks
(MobiHoc, 2001)
The Key to Bayesian Methods
P(A
∩
B)
P(AB) P(B)
P(BA) =  = 
P(A)
P(A)
This is Bayes Rule
Bayes, Thomas (1763)
An essay
towards solving a problem in the doctrine
of chances.
Philosophical Transactions of
the Royal Society of London,
53:370418
General Forms of Bayes Rule
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11/8/10
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General Forms of Bayes Rule
Based on Andrew L. Moore tutorials
An Example
Definition of Independent Events
Events E and F are called independent
iff P(E
∩
F)= P(E) P(F).
Note that this is equivalent to
P(EF)=P(E) and P(FE)=P(F).
Random Variables
•
A random variable is a function that maps a set of
outcomes to real numbers (or subsets of real numbers).
We use capital letters to represent random variables (Eg. X)
and lower case letters to represent particular outcomes.
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 Fall '08
 Myers
 Probability theory, random variable, Wireless Networks, N timeslots

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