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7/28/2019 RP1 Introduction
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Random ProcessesIntroduction
Professor Ke-Sheng Cheng
Department of Bioenvironmental SystemsEngineering
E-mail: [email protected]
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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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