Information Theory, Coding and Cryptography Unit-2 by Arun Pratap Singh

8/12/2019 Information Theory, Coding and Cryptography Unit-2 by Arun Pratap Singh

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UNIT IIINFORMATION THEORY, CODING & CRYPTOGRAPHY (MCSE 202)

PREPARED BY ARUN PRATAP SINGH 5/26/14 MTECH2nd SEMESTER


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PREPARED BY ARUN PRATAP SINGH 1

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STOCHASTIC PROCESS :

In probability theory, a stochastic process or sometimes random process (widely used) is a

collection ofrandom variables;this is often used to represent the evolution of some random value, or

system, over time. This is the probabilistic counterpart to a deterministic process (or deterministic

system). Instead of describing a process which can only evolve in one way (as in the case, for example,

of solutions of an ordinary differential equation), in a stochastic or random process there is some

indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely

many) directions in which the process may evolve.

In the simple case ofdiscrete time,as opposed to continuous time,a stochastic process involves

a sequence of random variables and the time series associated with these random variables (for

example, see Markov chain, also known as discrete-time Markov chain). Another basic type of a

stochastic process is arandom field,whose domain is a region ofspace,in other words, a random

function whose arguments are drawn from a range of continuously changing values. One approach to

stochastic processes treats them asfunctions of one or several deterministic arguments (inputs, in

most cases regarded as time) whose values (outputs) arerandom variables:non-deterministic (single)

quantities which have certain probability distributions. Random variables corresponding to various

times (or points, in the case of random fields) may be completely different. The main requirement is

that these different random quantities all have the same type. Type refers to the codomain of the

function. Although the random values of a stochastic process at different times may beindependent

random variables, in most commonly considered situations they exhibit complicated statistical

correlations.

Stock market fluctuations have been modeled by stochastic processes.

UNIT : II
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Given aprobability space and ameasurable space , an S-valued stochastic

process is a collection of S-valued random variables on , indexed by a totally

ordered set T("time"). That is, a stochastic processXis a collection

where each is an S-valued random variable on . The space S is then called the state

spaceof the process.

STATISTICAL INDEPENDENCE :

In probability theory, to say that two events are independent (alternatively called statistically

independentor stochastically independent)[1]means that the occurrence of one does not affect

the probability of the other. Similarly, tworandom variables are independent if the realization of one

does not affect the probability distribution of the other.

In some instances, the term "independent" is replaced by "statistically independent",

"marginally independent", or "absolutely independent"

For events :

Two events-

Two eventsAand Bare independent if and only if their joint probability equals the product of

their probabilities:
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.

Why this defines independence is made clear by rewriting withconditional probabilities:

and similarly

.

Thus, the occurrence of Bdoes not affect the probability ofA, and vice versa. Although the derived

expressions may seem more intuitive, they are not the preferred definition, as the conditional

probabilities may be undefined if P(A) or P(B) are 0. Furthermore, the preferred definition makes

clear by symmetry that whenAis independent of B, Bis also independent ofA.

More than two events

A finite set of events {Ai} is pairwise independentiff every pair of events is independent.[2]That

is, if and only if for all distinct pairs of indices m, n

.

A finite set of events is mutually independent if and only if every event is independent of any

intersection of the other events.[2]That is, iff for every subset {An}

This is called the multiplication rulefor independent events.

For more than two events, a mutually independent set of events is (by definition) pairwise

independent, but the converse is not necessarily true.

For random variables

Two random variables

Two random variablesXand Yare independentiff the elements of the-system generated by

them are independent; that is to say, for every a and b, the events {X a} and {Y b} are

independent events (as defined above). That is, X and Y with cumulative distribution

functions and , andprobability densities and , are independentif

and only if (iff) the combined random variable (X, Y) has ajoint cumulative distribution function

or equivalently, a joint density
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More than two random variables

A set of random variables is pairwise independent iff every pair of random variables is

independent.

A set of random variables is mutually independentiff for any finite subset and

any finite sequence of numbers , the events are

mutually independent events (as defined above).

The measure-theoretically inclined may prefer to substitute events {XA} for events {Xa} in

the above definition, where A is anyBorel set.That definition is exactly equivalent to the one

above when the values of the random variables arereal numbers.It has the advantage of working

also for complex-valued random variables or for random variables taking values in

anymeasurable space (which includestopological spaces endowed by appropriate -algebras).

Conditional independence

Intuitively, two random variablesXand Yare conditionally independent given Zif, once Zis known,

the value of Y does not add any additional information about X. For instance, two

measurements X and Y of the same underlying quantity Z are not independent, but they

are conditionally independent given Z(unless the errors in the two measurements are somehow

connected).

The formal definition of conditional independence is based on the idea ofconditional distributions.

IfX, Y, and Zarediscrete random variables,then we defineXand Yto beconditionally independent

givenZif

for allx, yand zsuch that P(Z= z) > 0. On the other hand, if the random variables arecontinuous and

have a jointprobability density functionp, thenXand Yareconditionally independent given Zif

for all real numbersx, yand zsuch thatpZ(z) > 0.

IfXand Yare conditionally independent given Z, then

for anyx, yand zwith P(Z= z) > 0. That is, the conditional distribution forXgiven Yand Zis the same

as that given Zalone. A similar equation holds for the conditional probability density functions in the

continuous case.

Independence can be seen as a special kind of conditional independence, since probability can be

seen as a kind of conditional probability given no events.
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Independent -algebras[edit]

The definitions above are both generalized by the following definition of independence for-algebras.

Let (,,Pr) be a probability space and let Aand Bbe two sub--algebras of .Aand Bare said to

be independentif, wheneverAAand BB,

Likewise, a finite family of -algebras is said to be independent if and only if for all

and an infinite family of -algebras is said to be independent if all its finite subfamilies are independent.

The new definition relates to the previous ones very directly:

Two events are independent (in the old sense)if and only if the -algebras that they generate

are independent (in the new sense). The -algebra generated by an eventE is, by

definition,

Two random variablesXand Ydefined over are independent (in the old sense) if and only

if the -algebras that they generate are independent (in the new sense). The -algebra

generated by a random variableX taking values in somemeasurable space Sconsists, by

definition, of all subsets of of the formX1(U), where Uis any measurable subset of S.

Using this definition, it is easy to show that if X and Y are random variables and Y is constant,

thenXand Yare independent, since the -algebra generated by a constant random variable is the

trivial -algebra {, }. Probability zero events cannot affect independence so independence also

holds if Yis only Pr-almost surely constant.

Properties :

Self-dependence

Note that an event is independent of itselfiff

.

Thus if an event or itscomplementalmost surely occurs, it is independent of itself. For example,

ifAis choosing any number but 0.5 from auniform distribution on theunit interval,Ais independent

of itself, even though,tautologically,Afully determinesA.

Expectation and covariance

IfXand Yare independent, then theexpectation operator Ehas the property
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and for thecovariance since we have

so thecovariance cov(X, Y) is zero. (The converse of these, i.e. the proposition that if two random

variables have a covariance of 0 they must be independent, is not true. Seeuncorrelated.)

Characteristic function

Two random variablesXand Yare independent if and only if the characteristic function of the

random vector (X, Y) satisfies

In particular the characteristic function of their sum is the product of their marginal characteristic

functions:

though the reverse implication is not true. Random variables that satisfy the latter condition are

calledsub-independent.

Examples :

Rolling a die

The event of getting a 6 the first time a die is rolled and the event of getting a 6 the second time

are independent. By contrast, the event of getting a 6 the first time a die is rolled and the event

that the sum of the numbers seen on the first and second trials is 8 are not independent.

Drawing cards

If two cards are drawn with replacement from a deck of cards, the event of drawing a red card on

the first trial and that of drawing a red card on the second trial are independent. By contrast, if two

cards are drawn without replacement from a deck of cards, the event of drawing a red card on

the first trial and that of drawing a red card on the second trial are again not independent.

Pairwise and mutual independence

Consider the two probability spaces shown. In both cases, P(A) = P(B) = 1/2 and P(C) = 1/4 The first

space is pairwise independent but not mutually independent. The second space is mutually

independent. To illustrate the difference, consider conditioning on two events. In the pairwise

independent case, although, for example, A is independent of both Band C, it is not independent

of BC:
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In the mutually independent case however:

See also for a three-event example in which

and yet no two of the three events are pairwise independent.

Pairwise independent, but not mutually independent, events.


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Mutually independent events.

BERNOULLI PROCESS :

In probability andstatistics,a Bernoulli process is a finite or infinite sequence of binary random

variables,so it is adiscrete-time stochastic process that takes only two values, canonically 0 and 1.

The component Bernoulli variablesXiare identical andindependent.Prosaically, a Bernoulli process

is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every

variable Xi in the sequence is associated with a Bernoulli trial or experiment. They all have the

same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be

generalized to more than two outcomes (such as the process for a six-sided die); this generalization

is known as theBernoulli scheme.

A Bernoulli processis a finite or infinite sequence ofindependentrandom variablesX1,X2,X3, ...,

such that

For each i, the value ofXiis either 0 or 1;

For all values of i, the probability thatXi= 1 is the same numberp.

In other words, a Bernoulli process is a sequence ofindependent identically distributedBernoulli

trials.

Independence of the trials implies that the process is memoryless. Given that the probability pis

known, past outcomes provide no information about future outcomes. (Ifpis unknown, however,

the past informs about the future indirectly, through inferences aboutp.)

If the process is infinite, then from any point the future trials constitute a Bernoulli process identical

to the whole process, the fresh-start property.

Interpretation

The two possible values of eachXiare often called "success" and "failure". Thus, when expressed

as a number 0 or 1, the outcome may be called the number of successes on the ith "trial".
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Two other common interpretations of the values are true or false and yes or no. Under any

interpretation of the two values, the individual variables Xi may be called Bernoulli trialswith

parameter p.

In many applications time passes between trials, as the index i increases. In effect, the

trials X1, X2, ... Xi, ... happen at "points in time" 1, 2, ..., i, .... That passage of time and the

associated notions of "past" and "future" are not necessary, however. Most generally,

anyXiandXjin the process are simply two from a set of random variables indexed by {1, 2, ..., n}

or by {1, 2, 3, ...}, the finite and infinite cases.

Several random variables and probability distributions beside the Bernoullis may be derived from

the Bernoulli process:

The number of successes in the first ntrials, which has abinomial distribution B(n,p)

The number of trials needed to get r successes, which has a negative binomial

distribution NB(r,p)

The number of trials needed to get one success, which has ageometric distribution NB(1,p),

a special case of the negative binomial distribution

The negative binomial variables may be interpreted as randomwaiting times.

Formal definition

The Bernoulli process can be formalized in the language of probability spaces as a random

sequence of independent realisations of a random variable that can take values of heads or tails.

The state space for an individual value is denoted by

Specifically, one considers thecountably infinitedirect product of copies of . It is

common to examine either the one-sided set or the two-sided

set . There is a naturaltopology on this space, called theproduct topology.The sets in

this topology are finite sequences of coin flips, that is, finite-lengthstrings ofHand T, with the rest

of (infinitely long) sequence taken as "don't care". These sets of finite sequences are referred to

as cylinder sets in the product topology. The set of all such strings form a sigma algebra,

specifically, a Borel algebra. This algebra is then commonly written as where the

elements of are the finite-length sequences of coin flips (the cylinder sets).

If the chances of flipping heads or tails are given by the probabilities , then one can

define a natural measure on the product space, given by (or

by for the two-sided process). Given a cylinder set, that is, a specific

sequence of coin flip results at times , the probability of observing

this particular sequence is given by
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where kis the number of times that Happears in the sequence, and n-kis the number of times

that Tappears in the sequence. There are several different kinds of notations for the above; a

common one is to write

where each is a binary-valued random variable. It is common to write for . This

probability Pis commonly called theBernoulli measure.[1]

Note that the probability of any specific, infinitely long sequence of coin flips is exactly zero; this

is because , for any . One says that any given infinite sequence

hasmeasure zero.Nevertheless, one can still say that some classes of infinite sequences of coin

flips are far more likely than others, this is given by theasymptotic equipartition property.

To conclude the formal definition, a Bernoulli process is then given by the probability

triple , as defined above.

BINOMIAL DISTRIBUTION :

Thelaw of large numbers states that, on average, theexpectation value of flipping headsfor any

one coin flip isp. That is, one writes

for any one given random variable out of the infinite sequence ofBernoulli trials that compose

the Bernoulli process.

One is often interested in knowing how often one will observe Hin a sequence of ncoin flips. This

is given by simply counting: Given n successive coin flips, that is, given the set of all

possiblestrings of length n, the number N(k,n) of such strings that contain koccurrences of His

given by thebinomial coefficient

If the probability of flipping heads is given by p, then the total probability of seeing a string of

length nwith kheads is

This probability is known as theBinomial distribution.
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Of particular interest is the question of the value of P(k,n) for very, very long sequences of coin

flips, that is, for the limit . In this case, one may make use ofStirling's approximation to

the factorial, and write

Inserting this into the expression for P(k,n), one obtains the Gaussian distribution; this is the

content of thecentral limit theorem,and this is the simplest example thereof.

The combination of the law of large numbers, together with the central limit theorem, leads to an

interesting and perhaps surprising result: the asymptotic equipartition property. Put informally,

one notes that, yes, over many coin flips, one will observe Hexactlypfraction of the time, and

that this corresponds exactly with the peak of the Gaussian. The asymptotic equipartition property

essentially states that this peak is infinitely sharp, with infinite fall-off on either side. That is, given

the set of all possible infinitely long strings of Hand Toccurring in the Bernoulli process, this setis partitioned into two: those strings that occur with probability 1, and those that occur with

probability 0. This partitioning is known as theKolmogorov 0-1 law.

The size of this set is interesting, also, and can be explicitly determined: the logarithm of it is

exactly theentropy of the Bernoulli process. Once again, consider the set of all strings of length n.

The size of this set is . Of these, only a certain subset are likely; the size of this set

is for . By using Stirling's approximation, putting it into the expression for P(k,n),

solving for the location and width of the peak, and finally taking one finds that

This value is the Bernoulli entropy of a Bernoulli process. Here, H stands for entropy; do not

confuse it with the same symbol Hstanding for heads.

von Neumann posed a curious question about the Bernoulli process: is it ever possible that a

given process isisomorphic to another, in the sense of theisomorphism of dynamical systems?

The question long defied analysis, but was finally and completely answered with the Ornstein

isomorphism theorem.This breakthrough resulted in the understanding that the Bernoulli process

is unique anduniversal;in a certain sense, it is the single most random process possible; nothing

is 'more' random than the Bernoulli process (although one must be careful with this informal

statement; certainly, systems that aremixing are, in a certain sense, 'stronger' than the Bernoulliprocess, which is merely ergodic but not mixing. However, such processes do not consist of

independent random variables: indeed, many purely deterministic, non-random systems can be

mixing).
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POISSON PROCESS :

Inprobability theory,a Poisson processis astochastic process that counts the number of event and

the time that these events occur in a given time interval. The time between each pair of consecutive

events has an exponential distribution with parameter and each of these inter-arrival times is

assumed to be independent of other inter-arrival times. The process is named after the French

mathematician Simon Denis Poisson and is a good model of radioactive decay,[1] telephone

calls[2]and requests for a particular document on a web server,[3]among many other phenomena.

The Poisson process is acontinuous-time process;the sum of aBernoulli process can be thought of

as its discrete-time counterpart. A Poisson process is a pure-birth process, the simplest example of

abirth-death process.It is also apoint process on the real half-line.

The basic form of Poisson process, often referred to simply as "the Poisson process", is a

continuous-timecounting process {N(t), t0} that possesses the following properties:

N(0) = 0

Independent increments (the numbers of occurrences counted in disjoint intervals are

independent of each other)

Stationary increments (the probability distribution of the number of occurrences counted in any

time interval only depends on the length of the interval)

Theprobability distribution of N(t) is aPoisson distribution.

No counted occurrences are simultaneous.

Consequences of this definition include:

The probability distribution of the waiting time until the next occurrence is an exponential

distribution.

The occurrences aredistributed uniformly on any interval of time. (Note that N(t), the total

number of occurrences, has a Poisson distribution over (0, t], whereas the location of an

individual occurrence on t(a, b]is uniform.)

Other types of Poisson process are described below.

1. Homogeneous

2. Non- Homogeneous
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Sample Path of a counting Poisson process

Homogeneous :

The homogeneousPoisson process counts events that occur at a constant rate; it is one of the

most well-knownLvy processes.This process is characterized by a rate parameter , also knownas intensity, such that the number of events in time interval (t, t + ] follows a Poisson

distribution with associated parameter . This relation is given as

where N(t+ ) N(t) = kis the number of events in time interval ( t, t+ ].

Just as a Poisson random variable is characterized by its scalar parameter , a homogeneous

Poisson process is characterized by its rate parameter , which is the expected number of

"events" or "arrivals" that occur per unit time.N(t) is a sample homogeneous Poisson process, not to be confused with a density or distribution

function.

Non-homogeneous :

A non-homogeneous Poisson process counts events that occur at a variable rate. In general, the

rate parameter may change over time; such a process is called a non-homogeneous Poisson

processor inhomogeneous Poisson process. In this case, the generalized rate function is

given as (t). Now the expected number of events between time aand time bis

Thus, the number of arrivals in the time interval (a, b], given as N(b) N(a), follows aPoisson

distribution with associated parameter a,b
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A rate function (t) in a non-homogeneous Poisson process can be either a deterministic function

of time or an independent stochastic process, giving rise to a Cox process.A homogeneous

Poisson process may be viewed as a special case when (t) = , a constant rate.

RENEWAL PROCESS :

Renewal theory is the branch of probability theory that generalizes Poisson processes for

arbitrary holding times. Applications include calculating the expected time for a monkey who is

randomly tapping at a keyboard to type the word Macbethand comparing the long-term benefits of

different insurance policies.

A renewal processis a generalization of the Poisson process. In essence, the Poisson process is

a continuous-time Markov process on the positive integers (usually starting at zero) which

has independent identically distributed holding times at each integer (exponentially distributed)

before advancing (with probability 1) to the next integer: . In the same informal spirit, we may

define a renewal process to be the same thing, except that the holding times take on a more general

distribution. (Note however that the independence and identical distribution (IID) property of theholding times is retained).

Let be a sequence of positive independent identically

distributedrandom variables such that

We refer to the random variable as the " th" holding time.

Define for each n> 0 :

each referred to as the " th"jump timeand the intervals

being called renewal intervals.

Then the random variable given by

(where is theindicator function)represents the number of jumps that have occurred by time t,

and is called a renewal process.
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Sample evolution of a renewal process with holding timesSiand jump times Jn.

The renewal equation

The renewal function satisfies :

where is the cumulative distribution function of and is the corresponding probability

density function.

Proof of the renewal equation :

We may iterate the expectation about the first holding time:

But by theMarkov property

So
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as required.

RANDOM INCIDENCE :

The Poisson process is one of many stochastic processes that one encounters in urban servicesystems. The Poisson process is one example of a "point process" in which discrete events(arrivals) occur at particular points in time. For a general point process having its zero tharrival attime T0 and the remaining arrivals at times T1, T2, T3, . . ., the interarrival times are

Such a stochastic process is fully characterized by the family of joint

pdf's for all integer values of p and all possiblecombinations of different n1, n2, . . ., where each n i is a positive integer denoting a particularinterarrival time. Maintaining the depiction of a stochastic process at such a general level,although fine in theory, yields an intractable model and one for which the data (to estimate all the

joint pdf 's) are virtually impossible to obtain. So, in the study of stochastic processes, one ismotivated to make assumptions about this family of pdf's that


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(1) are realistic for an important class of problems and(2) yield a tractable model.

We wish to consider here the class of point stochastic processes for which the marginal pdf's forall of the interarrival times (Yk) are identical. That is, we assume that

Thus, for Yk, if we selected any one of the family of joint pdf's fYn1,Yn2, . . ., Ynp(yn1, yn2, . . . , yk, . . .,

ynP) and "integrated out" all variables except yk, we would obtain fY(.). Note that we have said

nothing about independence of the Yk's

They need not be mutually independent, pairwise independent, or conditionally independent in

any way. For the special case in which the Yk's aremutually independent, the point process is

called a renewal process. The Poisson process is a special case of a renewal process, being

the only continuous-time renewal process having "no memory." However, the kind of process

we are considering can exhibit both memory and dependence among the inter-event times. In

fact, the dependence could be so strong that once we know the value of one of the Y k's we

might know a great deal (perhaps even the exact values) of any number of the remaining Yk's.

Example :

Consider a potential bus passenger arriving at a bus stop. The kth bus arrives Y ktime units after

the (k - 1)st bus. Here the Yk's are called bus headways. The probabilistic behavior of the Yk's will

determine the probability law for the waiting time of the potential passenger (until the next bus

arrives). Here it is reasonable to assume that the Yk's are identically distributed but not

independent (due to interactions between successive buses). One could estimate the pdf fY(.)

simply by gathering data describing bus interarrival times and displaying the data in the form of a

histogram. (This same model applies to subways and even elevators in a multielevator building.)

Suppose that buses maintain perfect headway; that is, they are always T0minutes apart. Then


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That is, the time until the next bus arrives, given random incidence, is uniformly distributed

between 0 and T0, with a mean E[V] = T0/2, as we might expect intuitively.

MARKOV MODULATED BERNOULLI PROCESS :

The Markov-Modulated Bernoulli Process (MMBP) model is used to analyze the delay

experienced by messages in clocked, packed-switched Banyan networks with kx koutput-

buffered switches. This approach allows us to analyze both single packet messages and

multipacket messages with general traffic pattern including uniform traffic, hot-spot traffic,

locality of reference, etc. The ability to analyze multipacket messages is very important for

multimedia applications. Previous work, which is only applicable to restricted message andtraffic patterns, resorts to either heuristic correction factors to artificially tune the model or

tedious computational efforts. In contrast, the proposed model, which is applicable to much

more general message and traffic patterns, not only is an application of a theoretically

complete model but also requires a minimal amount of computational effort. In all cases, the

analytical results are compared with results obtained by simulation and are shown to be very

accurate.


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DTMC - Discrete Time Markov Chains

IRREDUCIBLE FINITE CHAINS WITH APERIODIC STATES :

A Markov chain (discrete-time Markov chain or DTMC) named after Andrey Markov, is a

mathematical system that undergoes transitions from one state to another on a state space. It is

arandom process usually characterized asmemoryless:the next state depends only on the current

state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is

called theMarkov property.Markov chains have many applications asstatistical models of real-world

processes.
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DISCRETE TIME BIRTH DEATH PROCESS :

The birth

death process is a special case of continuous-time Markov process where the statetransitions are of only two types: "births", which increase the state variable by one and "deaths", which

decrease the state by one. The model's name comes from a common application, the use of such

models to represent the current size of a population where the transitions are literal births and deaths.

Birthdeath processes have many applications in demography, queueing theory, performance

engineering, epidemiology or in biology. They may be used, for example to study the evolution
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ofbacteria,the number of people with a disease within a population, or the number of customers in

line at the supermarket.

When a birth occurs, the process goes from state nto n+ 1. When a death occurs, the process goes

from state n to state n 1. The process is specified by birth rates and death

rates .

Example :

A pure birth processis a birthdeath process where for all .

A pure death processis a birthdeath process where for all .

A (homogeneous)Poisson process is a pure birth process where for all

M/M/1 modelandM/M/c model, both used in queueing theory,are birthdeath processes used to

describe customers in an infinite queue.

Use in queueing theory :

In queueing theory the birthdeath process is the most fundamental example of aqueueing model,

the M/M/C/K/ /FIFO(in completeKendall's notation)queue. This is a queue with Poisson arrivals,

drawn from an infinite population, and Cservers withexponentially distributed service time

with Kplaces in the queue. Despite the assumption of an infinite population this model is a good

model for various telecommunication systems.

M/M/1 queue

The M/M/1is a single server queue with an infinite buffer size. In a non-random environment the

birthdeath process in queueing models tend to be long-term averages, so the average rate of

arrival is given as and the average service time as . The birth and death process is aM/M/1 queue when,

Thedifference equations for theprobability that the system is in state kat time tare,
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M/M/c queue

The M/M/c is a multi-server queue with C servers and an infinite buffer. This differs from the

M/M/1 queue only in the service time, which now becomes

and

with

M/M/1/K queue

The M/M/1/K queue is a single server queue with a buffer of size K. This queue has applications

in telecommunications, as well as in biology when a population has a capacity limit. In

telecommunication we again use the parameters from the M/M/1 queue with,

In biology, particularly the growth of bacteria, when the population is zero there is no ability to

grow so,

Additionally if the capacity represents a limit where the population dies from over population,

The differential equations for the probability that the system is in state kat time tare,

MARKOV PROPERTY :

Inprobability theory andstatistics,the term Markov propertyrefers to the memoryless property of

astochastic process.It is named after theRussianmathematicianAndrey Markov.[1]

A stochastic process has the Markov property if theconditional probability distribution of future states

of the process (conditional on both past and present values) depends only upon the present state, not

on the sequence of events that preceded it. A process with this property is called aMarkov process.

The term strong Markov property is similar to the Markov property, except that the meaning of
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"present" is defined in terms of a random variable known as a stopping time.Both the terms "Markov

property" and "strong Markov property" have been used in connection with a particular "memoryless"

property of theexponential distribution.[2]

The term Markov assumptionis used to describe a model where the Markov property is assumed to

hold, such as ahidden Markov model.

A Markov random field[3] extends this property to two or more dimensions or to random variables

defined for an interconnected network of items. An example of a model for such a field is the Ising

model.

A discrete-time stochastic process satisfying the Markov property is known as aMarkov chain.

FINITE MARKOV CHAIN :
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CONTINUOUS-TIME MARKOV CHAIN :

In probability theory, a continuous-time Markov chain (CTMC[1] or continuous-time Markov

process[2])is a mathematical model which takes values in some finite or countable set and for whichthe time spent in each state takes non-negativereal values and has anexponential distribution.It is

acontinuous-time stochastic process with theMarkov property which means that future behaviour of

the model (both remaining time in current state and next state) depends only on the current state of

the model and not on historical behaviour. The model is a continuous-time version of the Markov

chain model, named because the output from such a process is a sequence (or chain) of states.
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A continuous-time Markov chain (Xt)t 0 is defined by a finite or countable state space S,atransition rate matrix Qwith dimensions equal to that of the state space and initial probability

distribution defined on the state space. For ij, the elements qijare non-negative and describe

the rate the process transitions from state ito statej. The elements qiiare chosen such that each

row of the transition rate matrix sums to zero.

There are three equivalent definitions of the process.[3]

Infinitesimal definition

LetXtbe the random variable describing the state of the process at time t, and assume that the

process is in a state iat time t. ThenXt+ his independent of previous values (Xs: st) and as h

0 uniformly in tfor allj

using little-o notation. The qij can be seen as measuring how quickly the transition

from itojhappens

Jump chain/holding time definition

Define a discrete-time Markov chain Yn to describe the nth jump of the process and

variables S1, S2, S3, ... to describe holding times in each of the states where the distribution ofSiis

given by qYiYi.Transition probability definition

For any value n= 0, 1, 2, 3, ... and times indexed up to this value of n: t0, t1, t2, ... and all states

recorded at these times i0, i1, i2, i3, ... it holds that

wherepijis the solution of theforward equation (afirst-order differential equation)
http://en.wikipedia.org/wiki/Transition_rate_matrixhttp://en.wikipedia.org/wiki/Continuous-time_Markov_chain#cite_note-norris1-3http://en.wikipedia.org/wiki/Continuous-time_Markov_chain#cite_note-norris1-3http://en.wikipedia.org/wiki/Continuous-time_Markov_chain#cite_note-norris1-3http://en.wikipedia.org/wiki/Little-o_notationhttp://en.wikipedia.org/wiki/Forward_equationhttp://en.wikipedia.org/wiki/First-order_differential_equationhttp://en.wikipedia.org/wiki/First-order_differential_equationhttp://en.wikipedia.org/wiki/Forward_equationhttp://en.wikipedia.org/wiki/Little-o_notationhttp://en.wikipedia.org/wiki/Continuous-time_Markov_chain#cite_note-norris1-3http://en.wikipedia.org/wiki/Transition_rate_matrix


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with initial condition P(0) is theidentity matrix.

HIDDEN MARKOV MODEL :

A hidden Markov model(HMM) is astatisticalMarkov model in which the system being modeled is

assumed to be aMarkov process with unobserved (hidden) states. A HMM can be considered the

simplest dynamic Bayesian network. The mathematics behind the HMM was developed by L. E.

Baum and coworkers.[1][2][3][4][5] It is closely related to an earlier work on optimal nonlinearfiltering

problem (stochastic processes) byRuslan L. Stratonovich,[6]who was the first to describe theforward-

backward procedure.

In simpler Markov models (like a Markov chain), the state is directly visible to the observer, and

therefore the state transition probabilities are the only parameters. In a hiddenMarkov model, the state

is not directly visible, but output, dependent on the state, is visible. Each state has a probability

distribution over the possible output tokens. Therefore the sequence of tokens generated by an HMM

gives some information about the sequence of states. Note that the adjective 'hidden' refers to the

state sequence through which the model passes, not to the parameters of the model; the model is still

referred to as a 'hidden' Markov model even if these parameters are known exactly.
http://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Markov_modelhttp://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Dynamic_Bayesian_networkhttp://en.wikipedia.org/wiki/Leonard_E._Baumhttp://en.wikipedia.org/wiki/Leonard_E._Baumhttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-1http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-1http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-3http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-5http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-5http://en.wikipedia.org/wiki/Filtering_problem_(stochastic_processes)http://en.wikipedia.org/wiki/Filtering_problem_(stochastic_processes)http://en.wikipedia.org/wiki/Ruslan_L._Stratonovichhttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-Stratonovich1960-6http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-Stratonovich1960-6http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-Stratonovich1960-6http://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithmhttp://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithmhttp://en.wikipedia.org/wiki/Markov_modelhttp://en.wikipedia.org/wiki/Markov_chainhttp://en.wikipedia.org/wiki/Markov_chainhttp://en.wikipedia.org/wiki/Markov_modelhttp://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithmhttp://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithmhttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-Stratonovich1960-6http://en.wikipedia.org/wiki/Ruslan_L._Stratonovichhttp://en.wikipedia.org/wiki/Filtering_problem_(stochastic_processes)http://en.wikipedia.org/wiki/Filtering_problem_(stochastic_processes)http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-5http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-3http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-3http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-1http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-1http://en.wikipedia.org/wiki/Leonard_E._Baumhttp://en.wikipedia.org/wiki/Leonard_E._Baumhttp://en.wikipedia.org/wiki/Dynamic_Bayesian_networkhttp://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Markov_modelhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Identity_matrix


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Hidden Markov models are especially known for their application intemporal pattern recognition such

as speech, handwriting, gesture recognition,[7] part-of-speech tagging, musical score

following,[8]partial discharges[9]andbioinformatics.

A hidden Markov model can be considered a generalization of a mixture model where the hidden

variables (or latent variables), which control the mixture component to be selected for each

observation, are related through a Markov process rather than independent of each other. Recently,

hidden Markov models have been generalized to pairwise Markov models and triplet Markov models

which allow to consider more complex data structures[10][11]and to model nonstationary data.
http://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Speech_recognitionhttp://en.wikipedia.org/wiki/Handwriting_recognitionhttp://en.wikipedia.org/wiki/Gesture_recognitionhttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-7http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-7http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-7http://en.wikipedia.org/wiki/Part-of-speech_tagginghttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-8http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-8http://en.wikipedia.org/wiki/Partial_dischargehttp://en.wikipedia.org/wiki/Partial_dischargehttp://en.wikipedia.org/wiki/Partial_dischargehttp://en.wikipedia.org/wiki/Bioinformaticshttp://en.wikipedia.org/wiki/Mixture_modelhttp://en.wikipedia.org/wiki/Latent_variableshttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-TMMEV-10http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-TMMEV-10http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-TMMEV-10http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-TMMEV-10http://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-TMMEV-10http://en.wikipedia.org/wiki/Latent_variableshttp://en.wikipedia.org/wiki/Mixture_modelhttp://en.wikipedia.org/wiki/Bioinformaticshttp://en.wikipedia.org/wiki/Partial_dischargehttp://en.wikipedia.org/wiki/Partial_dischargehttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-8http://en.wikipedia.org/wiki/Part-of-speech_tagginghttp://en.wikipedia.org/wiki/Hidden_Markov_model#cite_note-7http://en.wikipedia.org/wiki/Gesture_recognitionhttp://en.wikipedia.org/wiki/Handwriting_recognitionhttp://en.wikipedia.org/wiki/Speech_recognitionhttp://en.wikipedia.org/wiki/Time


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Information Theory, Coding and Cryptography Unit-2 by Arun Pratap Singh

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