I'm doing my PhD in geomechanics. I thought we use a Poisson-Weibull distribution (for the variability of a parameter at the rock), but reading more about the subject I think maybe is a Poisson-Weibull process and I don't know the difference. To complete the problem I'm not too knowledgeable about the language of mathematics, so if you could give me an example it would be awesome!
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Steffen Moritz
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user40948
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3The Wikipedia article on [stochastic processes](http://en.wikipedia.org/wiki/Stochastic_process) provides a clear, succinct, non-mathematical answer on the first line and right next to it is a picture of data that are typically thought of as resulting from a process. Have you investigated these materials? – whuber Feb 26 '14 at 16:13
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1A Poisson process is a model for a "real world" process that generates events in time. It has many distributions associated with it depending on what aspects of it you consider (e.g. the distribution of event-times in a fixed window is uniform, the distribution of time between events is exponential, the distribution of the number of events in a time interval is Poisson, etc). – Glen_b Feb 26 '14 at 23:24
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1Both of the comments could be considered answers, albeit brief answers. – Drew75 Feb 27 '14 at 07:44
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1Hi everyone, thanks for the answers! I have read the wikipedia article, but I think I have a more elementary problem to understand what real means "process" or even "distribution". I know the format of an exponetial distribution, or uniform, or Poisson, but I have a problem on understanding what this really means! I know that on a normal distribution we have a little chance to find events really "small" or really "big". I'm sorry, I'm sure your answers are good, but I still need a simpler explanation. Thanks anyway! – user40948 Feb 27 '14 at 12:46
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- Poisson distribution = a specific discrete probability distribution, i.e. a probability distribution characterized by a probability mass function. Specifically, in the case of Poisson, it is defined as $P(k \text{ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!}$, $\lambda \in \mathbb{R^+}, k \in \mathbb{N}$.
- 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 $(X_t)_{t\in T}$ defined on a probability space $(\Omega,\Sigma,P)\ ,$ where the set T is interpreted as ''time''. Specifically, a Poisson process can be defined in different ways, not necessarily using the Poisson distribution:
Franck Dernoncourt
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A random process is a sequence of random variables. That means, when talking about a process, e.g. Poisson process, an element of occurrences as a sequence in time is involved, while when we talk about random variables and their distribution, e.g. Poisson distribution, there is no such element involved, and we only have a random variable X with its associated distribution.
Example:
Random variable X: the number of phone calls to a receptionist per hour follows the Poisson distribution (and with the distribution known we can have the probability of receiving a certain number of phone calls in a given time interval);
Random process {X1, X2, ....}, where Xi: the time when the ith phone call was received, is a Poisson process.
bnd
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How does the Poisson process compare to the Gaussian process? (aka are kernels and conditioning methods used to infer what $y_i$ an unobserved $x_i$ might take on?) – jbuddy_13 Apr 07 '21 at 01:54