What is Random process: Definition and 23 Discussions

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

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  1. A

    Characterizing random processes

    Hello. I would like to kindly request some help with a multi-part problem on identifying random processes as an intro topic from my stats course. I’m fairly uncertain with this topic so I suspect my attempt is mostly incorrect, especially when specifying the parameters, and I would be grateful...
  2. A

    Poisson random process problem

    Hello all, sorry for the large wall of text but I'm really trying to understanding a problem from a study guide. I am quite unsure on how to approach the following multi-part problem. Any help would be appreciated. Problem: Useful references I'm using to attempt the problem My attempt: For...
  3. F

    I Random variable vs Random Process

    Hello, When flipping a fair coin 4 times, the two possible outcomes for each flip are either H or T with the same probability ##P(H)=P(T)=0.5##. Why are the 4 outcomes to be considered as the realizations of 4 different random variables and not as different realizations of the same random...
  4. C

    I Implementation of Correlated Gaussian Random Fields Model

    Hello everyone. I have been recently working in an optimization model in the presence of uncertainty. I have read https://www.researchgate.net/publication/310742108_Efficient_Simulation_of_Stationary_Multivariate_Gaussian_Random_Fields_with_Given_Cross-Covariance in which, a methodology for...
  5. C

    MATLAB Generation of a Gaussian a random process with Matlab

    Hello everyone. I am currently working with Matlab. I have a 2D gaussian kernel constructed using the muKL technique (first attached figure). I want to use it to generate realizations of a gaussian random process using the KL theorem. For that, I obtain then all eigenvectors and eigenvalues of...
  6. iVenky

    Power of noise after passing through a system h(t)

    **Reposting this again, as I was asked to post this on a homework forum** 1. Homework Statement Hi, I am trying to solve this math equation (that I found on a paper) on finding the variance of a noise after passing through an LTI system whose impulse response is h(t) X(t) is the input noise...
  7. A

    Simulating quantum coefficients

    Hi everyone, I want to generate 8 random variables (in reality to form 4 complex numbers) such that the sum of the 8 variables squared is equal to unity. The aim of generating such numbers is to perform a quantum simulation of 4 qubits (thus the 8 parameters). I've been trying to use...
  8. J

    Random process involving CDF and PDF of standard normal

    Homework Statement Let $$ \Phi(x)=\int_{-\infty}^{x} \frac{1} { \sqrt{2\pi} } e^{-y^2 /2} dy $$ and $$ \phi(x)=\Phi^\prime(x)=\frac{1} { \sqrt{2\pi} } e^{-x^2 /2} $$ be the standard normal (zero - mean and unit variance) cummulative probability distribution function and the standard normal...
  9. D

    Exploring the Expectation of a Random Process with Unequal Probabilities

    Hello, So I was asked this question the other day and don't really know how to go about it: "Define a random process by: Xn+1 = Xn+1 with probability 1/2 Xn+1 = 1/Xn with probability 1/2 Given that X0= 100, find the expectation of X10 " I've only ever really met random processes...
  10. DeathbyGreen

    Quantum fluctuations and order

    Is there a good explanation for how we can explain an ordered universe arising from an inherently uncertain quantum world? I'm aware of the conflict between special relativity and quantum vacuum fluctuations, but is this the only issue? The correspondence principle would seem to imply that...
  11. H

    True/False : Stationary process In stochastic process

    Stochastic process problem! 1. If Xn and Yn are independent stationary process, then Vn= Xn / Yn is wide-sense stationary. (T/F) 2. If Xn and Yn are independent wide sense stationary process, then Wn = Xn / Yn is wide sense stationary (T/F) I solve this problem like this: 1...
  12. H

    If X(t) is gaussian process, How about X(2t)?

    written as title, 1. If X(t) is gaussian process, then Can I say that X(2t) is gaussian process? of course, 2*X(t) is gaussian process 2. If X(t) is poisson process, then X(2t) is also poisson process?
  13. A

    Confusion about a random process

    Question already asked on http://math.stackexchange.com/questions/1310194/confusion-about-a-random-process?noredirect=1#comment2661260_1310194, but couldn't get an answer so reposting here...
  14. P

    Simulating random process (poisson process)

    Homework Statement I have a physical system, which I know the time average statistics. Its probability of being in state 1 is P1, state 2:P2 and state 3:P3. I want to simulate the time behavior of the system.Homework Equations N/AThe Attempt at a Solution I assume the rate of transition event...
  15. C

    Random process of uniform- graphing and pdf

    Homework Statement Word for word of the problem: Let N (t, a) = At be a random process and A is the uniform continuous distribution (0, 3). (i) Sketch N(t, 1) and N(t, 2) as sample functions of t. (ii) Find the PDF of N(2, a) = 2A. Homework Equations A pdf is 1/3 for x in...
  16. M

    "Understanding Random Process X(t) and Its Sample Realizations

    Problem statement: Define the random process X(t) = C where C is uniform over [-5,5]. a) Sketch a few sample realizations I need reassurance that if I do a a few sample realizations of this random process they are all going to look the same. They are going to be an horizontal line with...
  17. E

    The average of a random process

    Hello all, I have the following continuous-time random process: v(t)=\sum_{k=0}^{K-1}\alpha_k(t)d_k+w(t) where d_k are i.i.d. random variables with zero mean and variance 1, alpha_k(t) is given, and w(t) is additive white Gaussian process of zero-mean and variance N_0. Can we say...
  18. M

    Random process derived from Markov process

    I have a query on a Random process derived from Markov process. I have stuck in this problem for more than 2 weeks. Let r(t) be a finite-state Markov jump process described by \begin{alignat*}{1} \lim_{dt\rightarrow 0}\frac{Pr\{r(t+dt)=j/r(t)=i\}}{dt} & =q_{ij} \end{alignat*} when i \ne...
  19. J

    Discrete-time discrete-valued random process

    I have the random process: Sequence 1: 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 ... Sequence 2: 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 ... Sequence 3: 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 ... where let nature choose one of these sequences at random with equal probability 1/3. Can we say anything about this...
  20. Z

    C++ or verilog or random process ?

    C++ or verilog or random process ?? I'm currently in 3rd year of my undergrad in Electronics & Communication engg. For my 6th semester, I'm required to take 1 elective from the following- -Data Structures with C++ -Random Process -Digital System Design using Verilog -Analog & Mixed mode VLSI...
  21. lahanadar

    Random Process vs Random Variable vs Sample Space

    Hi everybody, I try to figure out connections and differences between random variables (RV), random processes (RP), and sample spaces and have confusions on some ideas you may want to help me. All sources I searched says that RP assigns each element of a sample space to a time function. I want...
  22. D

    N-dimensional RV vs DT Random process

    If a discrete random process can be viewed as a collection of random variables indexed by a value n and a discrete N dimensional random variable can be viewed as N random variables with with a joint pmf. In these cases it seems like there is not much difference between a N dimensional random...
  23. S

    How Do Random Signals Differ from Random Processes?

    What is the difference between Random Signal and Random Process? A random process does not necessarily imply and ensemble of random signals. Is this correct? M