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Random ProcessesIntroduction

Professor Ke-Sheng Cheng

Department of Bioenvironmental SystemsEngineering

E-mail: rslab@ntu.edu.tw

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Introduction

A random process is a process (i.e.,

variation in time or one dimensional

space) whose behavior is not completely

predictable and can be characterized by

statistical laws.

Examples of random processesDaily stream flow

Hourly rainfall of storm events

Stock index

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Random Variable

A random variable is a mappingfunction which assigns outcomes of arandom experiment to real numbers.Occurrence of the outcome followscertain probability distribution.Therefore, a random variable is

completely characterized by itsprobability density function (PDF).

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The term stochastic processes

appears mostly in statistical

textbooks; however, the termrandom processes are frequently

used in books of many engineering

applications.

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Characterizations of a

Stochastic Processes

First-order densities of a random processA stochastic process is defined to be completely ortotally characterized if the joint densities for the

random variables are known forall times and all n.

In general, a complete characterization ispractically impossible, except in rare cases. As a

result, it is desirable to define and work withvarious partial characterizations. Depending onthe objectives of applications, a partialcharacterization often suffices to ensure the

desired outputs.

)(),(),( 21 ntXtXtX

nttt ,,, 21

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For a specific t, X(t) is a random variablewith distribution .

The function is defined as the first-order distribution of the random variableX(t). Its derivative with respect to x

is the first-order density of X(t).

])([),( xtXptxF

),( txF

xtxFtxf

),(),(

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If the first-order densities defined for all timet, i.e. f(x,t), are all the same, then f(x,t) doesnot depend on tand we call the resultingdensity the first-order density of the randomprocess ; otherwise, we have a family of

first-order densities. The first-order densities (or distributions) are

only a partial characterization of the randomprocess as they do not contain information

that specifies the joint densities of the randomvariables defined at two or more differenttimes.

)(tX

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Mean and variance of a random process

The first-order density of a random process,

f(x,t), gives the probability density of therandom variables X(t) defined for all time t.The mean of a random process, mX(t), isthus a function of time specified by

ttttX dxtxfxXEtXEtm ),(][)]([)(

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For the case where the mean ofX(t) doesnot depend on t, we have

The variance of a random process, also a

function of time, is defined by

constant).(a)]([)( XX mtXEtm

2222 )]([][)]()([)( tmXEtmtXEt XtXX

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Second-order densities of a randomprocess

For any pair of two random variables X(t1)and X(t

2

), we define the second-order

densities of a random process as

or .

Nth-order densities of a random process

The nth order density functions forat times are given by

or .

),;,( 2121 ttxxf

),( 21 xxf

)(tXnttt ,,, 21

),,,;,,,(2121 nn

tttxxxf ),,,(21 n

xxxf

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Autocorrelation and autocovariancefunctions of random processes

Given two random variables X(t1) and X(t2),a measure of linear relationship between

them is specified by E[X(t1)X(t2)]. For arandom process, t1 and t2 go through allpossible values, and therefore, E[X(t1)X(t2)]can change and is a function oft1 and t2. The

autocorrelation function of a randomprocess is thus defined by

),()()(),( 122121 ttRtXtXEttR

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Stationarity of random processes

Strict-sense stationarity seldom holds for randomprocesses, except for some Gaussian processes.Therefore, weaker forms of stationarity areneeded.

nnnn tttxxxftttxxxf ,,,;,,,,,,;,,, 21212121

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.allforconstant)()( tmtXE

.andallfor,),(

21121221

ttttRttRttR

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Equality and continuity ofrandom processes

Equality

Note that x(t, i) = y(t,

i) for every

i

is not the same as x(t, i) = y(t, i) withprobability 1.

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Mean square equality

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