ECE 551 Probability and Stochastic Processes

1
ECE 551 Probability and Stochastic Processes

• Course director Xiangping (Sandra) Qin
• Required textbook
• Alberto Leon-Garcia, Probability and Random
  Processes for Electrical Engineering, Prentice
  Hall, 2nd edition (1994)
• Time Tuesday, Thursday
• Classroom A 0220
• Homework Thursday
• Office hours E0217, Wed 1100 1200am

2
Probability Models in Electrical and Computer
Engineering Probability

• ECE 551
• August 19, 2008

3
Motivation

• Application requirement
• Internet
• Computers, laptops and cell phones
• go on and off at random time
• Diverse applications
• Data email, web browsing
• Music Online radio, MP3 downloading and
  listening
• Movie Video on Demand
• Voice over IP phone calls over internet
• Internet 3D video gaming

4
Random Internet traffic
Time
Downloading
Music1
Music 2
Voice
Video
Data
5
Communication System
1
2
3
4
5
6
7
1
2
3
4
5
6
7
6
Toss a coin
H
T
H
H
T
T
H
Time
7
Modeling

• Deterministic Models
• Parameters determine outcome
• Circuit theory
• I V/R
• Probability Models
• Random experiment Outcome exhibit unpredictable
  variation and randomness
• Internet traffic, received packet over
  communication system, Coin tossing

8
Modeling

• Random outcome
• Sample space
• Internet traffic
• Observing Number of packets per second
• 1, 20
• Communication system
• Observing ratio of correctly received packets
• 0,1
• Coin tossing
• Observing Head or Tail
• Head, Tail

9
Statistical Regularity

• A model is used to predict
• Regularity in the random behavior
• Average obtained in long sequences of repetitions
  of random experiments consistently yields the
  same value
• Relative frequency
• Number of Heads / Number of experiments
• 0.5? ( Whats the condition )
• Probability of the outcome of head

10
Properties of the relative frequency

• Six-face Dice
• Six values 1,2,3,4,5,6
• N experiments
• N1 Number of times the outcome is 1 dot
• N2 , N3 , N4 , N5 , N6
• Relative frequency F1 N1 / N
• F2 , F3 , F4 , F5 , F6

11
Properties of relative frequency

• 0 Fi 1, 1 i 6
• ?i1..6 Fi 1
• Relative frequency of outcome is either 1 or 2
• Fa F1 F2
• Relative frequency for n random experiments
• fk(n) Nk(n) / n
• Limn -gt 8 fk(n) pk

12
Axiomatic Approach to a Theory of Probability

• Definitions
• A random experiment sample space S and possible
  outcomes
• Events subsets of S (event A)
• P(A) assigned to event A
• Axioms
• 0 PA 1
• PS 1
• If A and B are events that are mutually exclusive
  of each other PA or B PA PB

13
Example packet voice transmission system I

• Transmit 48 simultaneous conversations from city
  A and city B
• Speech of each speaker
• Voltage waveforms
• Digitalize and bundle into packets
• 10 ms speech ? one packet
• Do we need to transmit 48 packets per 10 ms?

14
Example packet voice transmission system II

• Speech speak and silence
• Silence Null packets(no need to transmit)
• Speak active packets
• Multiplexer
• Transmit M packets per 10 ms
• M lt 48
• Define A as the number of packets generated per
  10ms
• A lt M
• A gt M, Alt 48 discard A M packets

15
Example packet voice transmission system III

• Sample mean
• lt A gtn ?j1..n A(j) / n
• Expected value of A
• Limn -gt 8 lt A gtn EA
• EA ?k0..48 k p(k)
• Fraction of active packets that are discarded
• r ?kM1..48 (k-M) p(k) / ?k0..48 k p(k)
• M increases, r decreases

16
(No Transcript)