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Random process is the process in one direction which changes with time whose behavior of the variable cannot be predicted and can be characterized by statistical laws. Examples of random system are Daily stream flow , stock index , hourly rainfall of storm events . The characteristics of the random behavior is done by the probability distribution function .
Characterization of random process are
First order densities of a random process : It is defined by derivative of function F(x,t). F(x, t) does not depend on time . It will remains same for all the time . It does not contain any information that specifies the joint densities of random variable .
Second order densities of a random process : Second order densities are used to pair two random variable X(t1) and X(t2).
nth order densities of a random process : It is used for pairing nth random variable X(t1) , X(t2), X(t3) ,---------X(tn).
Autocorrelation and autocovariance function of a random process : Autocorrelation function is defined by R(t1,t2) =E[X(t1)X(t2)] = R(t2,t1) whereas Autocovariance function is defined by
C(t1,t2) = E [(X(t1)-mx(t1)(X(t2)-mx(t2)] = R(t1,t2)- mx(t1)mx(t2).
Application of random process are wide-sense stationary random processes, spectral representation. autoregressive moving average processes,analysis and processing of random signals through a linear system, cross-correlation and cross-spectrum .
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. There are various examples of random processes daily stream flow, hourly rainfall of storm events, stock index etc.
Random processes have the following properties:
• Random processes are functions of time.
• Random processes are random in the sense that it is not possible to predict exactly what waveform will be observed in the future.
• Suppose that we assign to each sample point ‘s’ a function of time with the label
X(t,s), = -T<t<T
And the sample function as
xj(t) = X(t,sj)
Random processes and LSI Systems are interrelated. It is point to be noticed when a random signal is processed by an LSI system, where x(n) and y(n) are random signals, and h(n) is a deterministic (i.e., non random) LSI system.
The input, x(n) is a random signal, so y(n) is, too. Random in, random out.
Two random processes are independent if the outcome of one does not influence the other. For example, rolling two dice. There are several reasons for the importance of random processes. Random variables and processes talk about quantities and signals which are unknown in advance. The data sent through a communication system is modeled as random variable.
Two random processes are dependent if the outcome of one can influence the other. Example: drawing a 5-card poker hand. The major topics to study in Random processes in systems are generalized harmonic analysis, signal processing applications, types of random processes, autocorrelations, Gaussian processes, Markov processes, Stationary processes etc.
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