Currently I am reading about stochastic processes. I cannot understand how one can extend distribution to a process (like a particular example is Poisson distribution and Process). But can we extend the same idea for other distributions like weibull distribution etc. What are the sufficient and necessary conditions to get process from distribution?
I searched about Poisson distribution and process and i reached at the following conclusion.
A Poisson process is a non-deterministic process where events occur continuously and independently of each other. An example of a Poisson process is the radioactive decay of radionuclides.
A Poisson distribution is a discrete probability distribution that represents the probability of events (having a Poisson process) occurring in a certain period of time.
Is there is any reference how we can extend distribution to a process especially for continuous families?
You are confusing things. All stochastic process are defined basically from a probability distribution. They are "random variables" depending on a parameter (it can be continuous or not). A distribution assigns a "likelihood to occur" to the image values of a process. You should check the Papoulis' introductory book on stochastic processes.
There are an excellent book about the on this topic:
Stochastic Processes, 2nd Edition
Sheldon M. Ross
The Poisson process is an important random process in probability theory. It is used to model random points in time and space, such as the times of radioactive emissions, the arrival times of customers at a service center, and the positions of flaws in a piece of material. Several important probability distributions arise naturally from the Poisson process--the Poisson distribution, the exponential distribution, and the gamma distribution. Piere Pudio has explained in the simplest possible way. Poisson process is about counting the number of events in a given interval.
"Adventures in Stochastic Processes", S. Resnick, read the section "the distribution of a process"
First, a random process is a sequence of random variables, which models occurrence of random events. The Poisson process as a random process models random events in time and space.
A typical example are arrivals of email messages in the email server: T1, T2, ..., where Ti is the time when i-th email arrives, and DT_i = Ti-T{i-1} is the inter-arrival time. If we assume that T1, T2, T3, .... is a Poisson process (i.e. it satisfy property of homogeneity and independence), then DT_i has exponential distribution, Ti has Gamma distribution (since it is the sum of i DT_i: Ti=T1+DT_1+...+DT_{i-1}), and if we denote with Nt the number of Ti
Probability distribution is a characteristics of a stochastic process. Stochastic process is defined by the transition probability distribution from a given state (\ksi_1) to another state (\ksi_2) in a given time interval.
In other words, it describes probability to observe a certain process state (\ksi_2) some "time" (\tau) later, provided it is at (\ksi_1) now.
Think of brownian particle at location (\ksi_1) now, if it's diffusion coefficient is D, probability to observe it at (\ksi_2) after time \tau will be proportional to exp(|\ksi_2 - \ksi_1|^2/(D*tau)).
If the process is stationary, transition distribution does not depend on time, but only on the state variables, and delay time (\tau), integrating over all possible transitions one can find distribution function describing probability to observe a given state (\ksi) at any time.
Hope I did not confuse you even more.
I want to know the difference between stochastic process and distribution function. I read answers given above. but these answers are little bit confusing. May you tell me basic difference between them.
Poisson distribution = a specific discrete probability distribution, i.e. a probability distributioncharacterized by a probability mass function. Specifically, in the case of Poisson, it is defined as P(k events in interval)=λke−λk!P(k events in interval)=λke−λk!, λ∈R+,k∈Nλ∈R+,k∈N.but Poisson process = a stochastic process, i.e. a collection of random variables representing the evolution of some system of random values over time. In other words, it is a family of real random variables (Xt)t∈T(Xt)t∈T defined on a probability space (Ω,Σ,P) ,(Ω,Σ,P) , where the set T is interpreted as ''time''.
so, Specifically, a Poisson process can be defined in different ways, not necessarily using the Poisson distribution
This link is useful
https://stats.stackexchange.com/questions/87988/what-is-the-difference-between-a-distribution-and-a-process-poisson/285624
Hope you good luck
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