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Statical Parameters

• Parameters are numbers that summarize data for an entire population

• Statistics are numbers that summarize data from a sampleExample• A researcher wants to estimate the average height of

women aged 20 years or older. From a simple random sample of 45 women, the researcher obtains a sample mean height of 63.9 inches

Random /Stochastic process

• random process, is a collection of random variables, representing the evolution of some system of randomvalues over time.

Types of Random Process(a)continuous time continuous state (b) continuous time discrete state (c) discrete time continuous state (d) discrete time discrete state

Random /Stochastic process

Discrete Random process• In telecommunication we deal with discrete-time

random process • A discrete-time random process x(n) is a collection,

or ensemble, of discrete-time signals• A discrete state stochastic process is often called a

chain

Discrete Random process• statistical properties of a random process may be obtained

in two waysTime Average• The average determined by measurements on a single

sample function at successive times will yield a time average

Ensemble average• The statistical average made at some fixed Time on all the

sample functions of the ensample is the ensemble average

label of components x1, x2….x10 and the times t1, t2…t10 the numbers for the ten components at any one time (i.e. a vertical column) add them and divide by ten is Ensemble Average. and average the voltages at the ten different times is Time Average

Ergodicity & Stationary processes

Ergodicity• When the time average of a process is equal to the

ensemble average, it is said to be ergodicStationary• the statistics do not change with time. The behavior is

time-invariant, even though the process is random.

• Telephone traffic is nonstationary. But the traffic obtained during busy hour may be considered as stationary

Pure Chance traffic• pure-chance traffic means that call arrivals and terminations are

independent• user may make a call at any time of the day• number of sources from which calls originate is infinite• All the sources originated calls simultaneously there would be no

liklihood of any further calls occurring.• Propotion sources to the number of calls in progress at any one

time is usually very small• it is reasonable to assume that subscribers' traffic is originated

on a basis of pure chance.

Markov process• Markov process is developed by A.A. Markov on 1907• It can be used to model a random system that changes

states according to a transition rule that only depends on the current state

• (a.) The number of possible outcomes or states is finite.• (b.) The outcome at any stage depends only on the

outcome of the previous stage.• (c.) The probabilities are constant over time.• It is also called markov chains

Continuous-time Markov chain & Discrete-time Markov chain • A continuous-time Markov chain is one in which changes to the system can

happen at any time along a continuous intervalExample: number of cars that have visited a drive-through at a local fast-food

restaurant during the day• A discrete-time Markov chain is one in which the system evolves through

discrete time steps. So changes to the system can only happen at one of those discrete time values

Example This is discrete because changes to the system state can only happen on

someone's turn.

Birth and death process• Every incoming call request is consider as birth • Every user that after being served leave the system

is considered as death. • transition from n to n-1 if any ended condition

(death)• n to n+1 if any just occur condition (birth).

•P(k) is the probability of state k• λk is called the birth rate in state k.•probability of transition from state k to state k– 1 in the time interval ∆t is µk ∆t where µk is called the death rate in state k•The probability in ∆t, from state kto a state other than k+ 1 or k– 1 is zero