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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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Topics for Random Processes in Systems  Assignment  help : 

  • Probability and random variables,Axioms and properties of probability,Conditional probability,Random variables,Density functions,Normal random variables,Two random variables,Joint density and computation of probability,Independence, correlation,Linear mean square estimation,Random sequences,Conditional densities, Chapman-kolmogorov equation,Normal sequences, sample mean,Markov and chebychev inequalities,Convergence of sequences,Laws of large numbers,Central limit,Random processes
  • Autocorrelation,Autocovariance,Strict sense, Wide sense & stationary increments,Power spectral density,Relation to fourier transform,Discrete-time vs continuous-time,White noise, Spectral estimation,Response of linear systems to random inputs,Time doman analysis,Mean and auto correlation of output,Cross correlation of input with output,Frequency domain analysis,Bandpass signals and filters,Ergodicity,Mean ergodicity,Correlation and distribution ergodicity,Expansions of random processes,Sampling
  • Karhunen-loeve,Markov processes,Poisson revisited,Queues,Discrete-time,Discrete-state,Homogeneity, Reducibility, Recurrence,Continuous-time, Discrete-state,Diffusion equations,Simulation of random processes,Mean square estimation,Orthogonality principle for n observations,Whitening,Linear and nonlinear estimation,Rank reduction,Continous- time observations, wiener filter,Probability spaces and models ,Probability axioms,Probability models,Conditional probability

Complex topics covered by Random Processes in Systems Assignment online experts :

  • Total probability and bayes rule,Independence discrete random variables ,Single random variable ,Multiple random variables,Bernoulli random process,Conditioning,Covariance,General random variables,Single random variable ,Multiple random variables,Probability spaces, conditional probability,independence,Multiple rvs
  • Joint distribution,Bounds, convergence notions and limit theorems,Markov chains,Chapman-kolmogorov equations,Classes of states,Irreducible markov chains,Limiting distributions,ranking of nodes in graphs,Exponential times, memoryless property, Counting processes,Poisson processes, interarrival times,Continuous-time markov chains, Birth and death processes ,Transition probability function, Kolmogorov’s equations, Ergodicity, balance equations ,Queuing theory , Queue tandem,Brownian motion,Geometric brownian motion
  • White noise ,Black-scholes formula for options pricing ,Stationary processes, Implications,Wide-sense stationarity,Linear filtering of wide-sense stationary processes,Matched filter,Wiener filter,Mmse estimation,Derived distributions ,Moment generating function ,Jointly gaussian random variables ,Discrete-time random processes ,Wide-sense and strict-sense stationarity,Autocorrelation and power spectral density,Transfer of random processes through linear systems,Gaussian processes,Continuous-time gaussian processes ,Markov chains

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