mth263 lecture 1

MTH 263 Probability and Random Variables

Lecture 1Dr. Sobia Baig

Electrical Engineering DepartmentCOMSATS Institute of Information Technology, Lahore

ContentsContents

• Probability and Random Variables‐briefProbability and Random Variables brief introduction/motivation

• Mathematical Models as tools in Analysis and• Mathematical Models as tools in Analysis and Design

D t i i ti M d l– Deterministic Models

– Probability Models

• Examples

Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 2

A Typical Communication SystemA Typical Communication System

• Probabilistic methods in making decisions about the transmitted/received message / g


Digital Communication with b b l fProbability of Error


Probability/ Uncertainty Element in Systems

• Wireless communication networks provide voice e ess co u cat o et o s p o de o ceand data communications to mobile users in severe interference environments.

• The vast majority of media signals, voice, audio, images, and video are processed digitally.

• The systems the designers build are unprecedented in scale and the chaotic environments in which they must operate areenvironments in which they must operate are untrodden terrritory.

• So there is “Uncertainty”So there is Uncertainty


Probability ModelsProbability Models

• Probability models are one of the tools thatProbability models are one of the tools that enable the designer to make sense out of the chaos and to successfully build systems thatchaos and to successfully build systems that are efficient, reliable, and cost effective


MATHEMATICAL MODELS AS TOOLS IN ANALYSIS AND DESIGNAND DESIGN

• Experiments: Costly way of testing a design or solve a problem.

• Model: Approximate representation of a physical situation. • Useful Model: Able to explain all relevant aspects of a givenUseful Model: Able to explain all relevant aspects of a given

phenomenon. • Mathematical Models: If observational phenomenon has

measurable properties then a mathematical modelmeasurable properties then a mathematical model consisting of a set of assumptions about the system is employed

• Conditions under which an experiment is performed and aConditions under which an experiment is performed and a model is assumed are very critical. Change the assumptions then a “good” model can be a great failure.


Computer Simulations & Deterministic d lModels

• Computer Simulation Models: They mimic or p ysimulate the dynamics of a system

• Deterministic Models: Lab and textbook cases, conditions determine outcomeconditions determine outcome

• 1. Circuit Theory • 2 Ohm’s Law• 2. Ohm s Law • 3. Kirchoffs’ Laws • 4 Transforms: FFT; Laplace Transforms4. Transforms: FFT; Laplace Transforms • 5. Convolution :Input/output behavior of systems with well‐defined coefficients


Probability ModelsProbability Models

• Probabilistic (Stochastic Random) Models:Probabilistic (Stochastic, Random) Models: involve phenomena that exhibit unpredictable variation and randomnessvariation and randomness.

h h b ll (Ex: Urn with three balls; (0,1,2 marked) ‐‐ Outcome: A number from the set {0,1,2} { , , }‐‐ Sample Space: All possible outcomes of an experiment: S = {0,1,2}


Statistical RegularityStatistical Regularity

• Relative Frequency:Relative Frequency:


Properties of Relative FrequencyProperties of Relative Frequency

• Suppose a random experiment has K possibleSuppose a random experiment has K possible outcomes: S = {1,2,…,K}. Then in “n” trials we havehave

Ex: Consider the 3‐ball urn experiment A: even = {0,2} then,


• Disjoint (mutually exclusive) events: If A or BDisjoint (mutually exclusive) events: If A or B can occur but not both, then

• Relative frequency of two disjoint events is h f h i i di id l l i fthe sum of their individual relative frequency.

•


Axioms of ProbabilityAxioms of Probability

• Kolmogorov’s axioms to form a Theory of Probability: Assumptions:

1. Random experiment has been defined and the sample space S has been identified.

2. A class of subsets of S has been specified. 3. Each event A has been assigned a number P(A) such that,

– 1 0 ≤ P(A) ≤ 11. 0 ≤ P(A) ≤ 1 – 2. P(S) = 1 – 3. If A and B are mutually exclusive events then

• P(A or B) = P(A) + P(B)P(A or B) = P(A) + P(B)

• Kolmogorov’s axioms are sufficient to build a consistent Theory of ProbabilityTheory of Probability.


ExampleExample

• Example: Packet Voice CommunicationExample: Packet Voice Communication system Efficiency

• Due to silences voice communication is veryDue to silences voice communication is very inefficient on dedicated lines. It is observed that only “1/3” of the time actual speech goes through. How to increase this rate by using prob. approaches???

• Solution: Error vs rate trade off in digital information (BCS) transmission/storage


Packet Voice Communication system ff ( )Efficiency (2)



• A is the outcome of a random experiment, determining which p , gpackets contain active speech

• M<48 packets are transmitted every 10 ms

• If A≤ M, then all packets are active

• If A> M, then A‐M active packets are discarded randomly



• E[A]=48*1/3E[A]=48 1/3

• E[A]=16


ExampleExample


Example: Signal Enhancement Using lFilters

• Given a signal x(t) corrupted with noise andGiven a signal x(t) corrupted with noise and has a Signal‐to‐Noise Ratio value SNR. If you filter this noisy signal with a properly designedfilter this noisy signal with a properly designed adaptive filter to suppress noise, we obtain an enhanced signal (smoothed by the filter )enhanced signal, (smoothed by the filter.)


Example: Signal Enhancement Using lFilters


Resource SharingResource Sharing

• Example: Multi User Systems with Queues:Example: Multi User Systems with Queues: Resource sharing


Multi User Systems with Queues: hResource sharing


System Reliability: Cascade vs. Parallel Systems

• Issues: Need of a clock vs the system delay orIssues: Need of a clock vs. the system delay or throughput rate.


ProblemProblem


Reading AssignmentReading Assignment

• Text Book Chapter 1 pages 17 ‐34Text Book, Chapter 1, pages 17 34


SummarySummary

• Mathematical Models– Relate system parameters and variables– Allow system designers to predict system performance by using

equations– Experimentation may be too costly– Experimentation may be too costly– Experimentation may not be feasible

• Computer simulation models predict system performance• Deterministic models give output of an experiment with an exact• Deterministic models give output of an experiment with an exact

outcome• Probability models determine probabilities of the possible

outcomes• Probabilities and averages for a random experiment can be found

experimentally by computing relative frequencies and sample averages in a large number of trials of experiment


mth263 lecture 1

Documents

h h b ll

contents probability

random experiment

silences voice communication

example example

pb paorb

useful model

ohm slaw