probabilty theory

8/13/2019 Probabilty theory

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Why do we need to study Probabilityand Random Processes

Probability theory provides a power tools to Explain Model Analyze Design

the technologies which are developed.


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PROBABILITY THEORY and PROBABILITYMODELs provide a tool that enables thedesigner to successfully design systems thanmust operate in RANDOM ENVIORNMENT, butnevertheless are efficient, reliable and costeffective.


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Examples

Signal Processing Optical Communication Wireless Communications Variability in electronic circuits. Computer network traffic.


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Radar Communication

Signal hidden in Noise.Noise is a random signalNoise can be characterized byits probabilitistic nature.


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Optical Communication System Photo detector acts as the interface between optical and electronic subsystem. The number of photoelectrons produced by the photo detector depends on the

intensity of light and is modeled by POISSON RANDON VARIABLE. In deciding whether the transmitted bit is zero or one, the receiver counts the

number of photoelectrons and compares it with threshold. System performance is determined by computing the probability that the

threshold is exceeded.


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Wireless Communication System

In order to increase the weak signals we useAmplifiers.

These amplifiers generate thermal noise,which is added to the desired signal.

Noise has a Guassian distribution (UnderlyingPhysics)


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Variability in electronics circuits

Although circuit manufacturing processattempt to ensure that all items have nominalparameters values, there is always someamount of variations among items.

Using probability theory we can Estimate the average values in a batch of items

without testing all of them. Check how good is our estimate


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Reliability of Systems Two basic Configurationss

Series System If one component fails the entire system fails.

Parallel System

If one component fails still the system functions It is not possible to predict exactly when a component

will fail However probability theory allows us to evaluate the

MEASURES OF RELIABILITY. Average time to failure. Probability that a component is still functioning after

certain time has elapsed.


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Markov Chain / Markov Process

All digital communication sytems are modeledas markov process/ markov chain.

Digital circuits with internal memory such asflipflops, RAM, registers, Finite state machinesare modelled by markov process / markovchain.


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Queing Theory

The question that usually arises in acommunication system is queing problem.

Que : buffer which stores messages. Assume server can process one message at a

time. So if more than one message is being

processes then since the buffer has limitedspace the other messages have to wait forservice.


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Performance measures to be found Average number of customers (messages) in the

system (L) Average number of customers waiting in que. (Lq) Average number of customers in service (Ls) L =Lq+ Ls Average time customer spends in que (Wq)

Average time a customer spends in service (Ws) Average time a customer spends in system (W). W= Wq + Ws


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Deterministic Experiment

An experiment whose outcome or result canbe predicted with CERTAINITY is calleddeterministic experiment.

For example: If E is the potential differencebetween the two ends of a resistor (R) , thenthe current flowing through the resistor isuniquely determined by Ohms law: I= E/R

This is a Deterministic experiment.


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Random Experiment

Any experiment in which ALL the outcomesare known in ADVANCE but the outcome of aPARTICULAR PERFORMANCE of theexperiment cannot be predicted with certaintyis called as Random experiment.


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Example of Random Experiment

Tossing a coin: Two possible outcomes (Head or Tail) But outcome of a particular toss cannot be

predicted. A fair 6 face cubic dice is rolled.

It is known that the outcome will be any of the 6possible values

But it cannot be predicted what exactly theoutcome will be, when the dice is rolled at a pointof time.


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Example of Random Experiment

Although the number of telephone callsreceived in a board in a 5 min interval is a non negative interger

we cannot predict exactly the number of callsin the next 5 mins.

In such cases we talk of CHANCE or thePROBABILITY (RELATIVE FREQUENCY) ofoccurrence of a particular outcome


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Sample space

We perform an experiment which can have anumber of different outcomes.

The sample space is the set of all possibleoutcomes of the experiment. We usually call itS.


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Sample Space / SET

Set A contains elements a 1, a 2,an. Element a k which is an element of A can be represented as:

Element a k which is not an element of A can be represented as:


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Example

It is important to be able to list the outcomesclearly.

For example: if I plant ten bean seeds and count the number

that germinate, the sample space is

S = {0,1,2,3,4,5,6,7,8,9,10}.

If I toss a coin three times and record the result,the sample space is

S = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT},


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Examples of Sample Space


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Example of Sample Space


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Sample space for variability inelectronic circuits


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Sample space forComputer network traffic


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Noise voltage is modelled by a sample spaceconsisting of all real numbers,

i.e., = (,). Outcomes: the individual numbers such as

1.5, 8, and are outcomes. Events: Subsets such as the interval

[0,5] = {v : 0 v 5} Another event would be {2,4,7.13}


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Event Elements or points in the sample space are called as OUTCOMES. Collection of outcomes is called : Event Event is a subset of sample space S. Event is a collection of certain sample points of S. Example: If I toss a coin three times and record the result, the sample space is

S = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}, let A be the event more heads than tails and B the event heads on last throw. Then

A = {HHH,HHT,HTH,THH}, B = {HHH,HTH,THH,TTH}.


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Singleton Set

Singleton sets, that is sets consisting of a single point, are also events; e.g., {1.5}, {8}, {}

Be sure you understand the difference between the outcome 8 and the event {8}, which is the set

consisting of the single outcome 8.


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EVENT SAMPLE SPACE / SET which consists of all possible

outcomes of the experiment. If the sample space is the set of all triples ( b1,b2,b3),

where the bi are 0 or 1, then any particular triplet, say (0,0,0) or (1,0,1) would be

an outcome. An event would be a subset of the sample space

(universal set). such as the set of all triples with exactly one 1; i.e., {(0,0,1), (0,1,0), (1,0,0)}.

An example of a singleton event would be {(1,0,1)}.

EVENT


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EVENT In modeling the resistance and capacitance of the RC filter

above,

we suggested the sample space = {(r,c) : 95 r 105 and 300 c 340}.

OUTCOME: If a particular circuit has R=101 ohms and C =327 F ,

this would correspond to the outcome (101,327), which isindicated by the dot in Figure. EVENT

If we observed a particular circuit with R97 ohms andC

313 F , this would correspond to the event {(r,c) : 95 r 97 and 313 c 340}, which is the shaded region in Figure

The dot is the outcome (101,327). The shaded region is the

event {( r,c ) : 95 r 97 and 313 c 340}.


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EVENTS

Exhaustive events Favorable events Mutually exclusive events Equally likely events Independent events


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Venn Diagram Sets can be represented graphically by means of

Venn diagram. We first define a universal set S is the universal set

S={x: x is all positive integers} A subset (EVENT) A can be represented by means

of venn diagram as shown in fig. A={1,2,3,4,5,6} Then its venn diagram is as follows:


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Venn diagram

S={x: x is all positive integers} A={1,2,3,4,5,6}


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Equality of Sets

Two sets A and B. They are said to be equal if That is every element of B is contained in A

and every element of A is contained in B. That is sets A and B contain exactly same

elements.


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Example

S={x: x is all positive integers} Set A={1,2,3,4,5,6} Subset set B = {1,3,5} B is a subset of A. Whereas set C = {x:x is a positive integer 6}

and set D = {2,4,6} C and D are same since both contain the same

elements


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Venn diagram of above example


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Empty Set

A empty set is a set that contains no elements. It is denoted by NOTW: The set A={0} is not an empty set

since it contains the element 0.


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Set Operations / MathematicalOperations

We can build new events from old events byusing set operators.

Union Intersection Complement


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Union

Union of two sets A and B is a third set Cwhich includes all of the elements that are ineither A or B or both.


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Union


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Intersection

The intersection of two sets, A and B, is theset of all elements that are in both A and B."


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Intersection


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Complement of an event


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Summary


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Disjoint or Mutually Exclusive Events


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Mutually exclusive set / event

In throwing a dice, the outcomes are mutually exclusive as occurrenceof one face excludes the occurrence of remaining 5 faces.


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Exhaustive Event

All possible outcomes of an experiment iscalled exhaustive event.

Example: In tossing a coin either {head} or {tail} turns up. There is no other possibility Therefore these two outcomes are exhaustive

events.


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Favorable Event

The number of outcomes which results to thedesired event are called favorable event.

Example: In throwing of 2 dice , the number of cases

favorable to get a sum of 6 is: (1,5); (2,4); (3,3); (4,2); (5,1) i.e there are 5 favorable cases.


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Equally likely event Two or more events are said to be equally likely if

the chances of their happening is equal. That is there is no preference of occurance of one

event over the other. Example:

In throwing a coin , getting a {head} or a {tail} isequally likely .

In throwing a dice getting events as {1}, {2}, {3}, {4},{5}, {6} are equally likely

Such events are called as equally likely events.

probabilty theory

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