Talk:Bernoulli process

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problems

formal definition is not very formal. in fact, it doesn't state the independence at all. bad.

next, bernoulli process are usually not even required to be identically distributed, just independent.

Is the Bernoulli process a sequence of random variables or a single one whose domain is the sequences in a two-element set, canonically {0,1}? The introduction and informal definition clearly say the former but the definition of Bernoulli sequence Zn implies the latter. One element omega in the sample space corresponds to the entire random sequence one sequence <X0, X1, ...>, ie the B process rather than one B trial Xi.

Either way, the formal definition seems bad to me. Does "for every omega in Omega" belong here, quantifying a probability statement, "the probability of Xi(omega)=1 with probability p ...". For every omega, Xi=1 or Xi=0 without any probability. Right? P64 (talk) 21:55, 18 February 2010 (UTC)[reply]

it is a discrete-time Lévy process. seems to be completely false : it would require that $X_{4}-X_{3}$ , $X_{3}-X_{2}$ , $X_{2}-X_{1}$ to be independant ; for instance, $(X_{3}-X_{2},X_{2}-X_{1})=(1,1)$ should have probability 1/16, while clearly the probability is 0 for $X_{3}-X_{1}=2$ never happens. I erase any mention of Levy processes in the page, until I am proved wrong. Hope it won't happen :-) --Chassain (talk) 18:05, 1 January 2010 (UTC)[reply]

older

I believe an equation associated with this is:

 tCw * (p(s)^w) * (p(f)^(t-w))

Where t is total number of trials, w is the number of successes wanted, p(s) is the probability of success, p(f) is the probability of failure. Is this correct, and perhaps added? I am not sure. -- KneeLess 04:09, 20 Sep 2004 (UTC)

That is the probability mass(weight,density,measure) at w for the binomial distribution with t trials and p(s) probability. Yes, it is reasonable to say that is all about the Bernoulli process. The article now permits a finite process. If the process must be infinite then everything binomial may be interpreted in terms of its finite subsequences such as the first t trials.

How to present this point will follow from answers to the more general winter 2010 discussion. --P64 (talk) 21:26, 22 February 2010 (UTC)[reply]

technical

The mathematics articles in Wikipedia seem as if they are all written for somebody with a mathematical or engineering background. I thought the wikipedia was written for general readers? 69.140.164.142 05:20, 27 March 2007 (UTC)[reply]

Each article is written at a level appropriate to the subject. There is no point, for example, in writing an article on motives for the general reader. However, in this case, I agree with you: an article on an important process like this should at least start in a way which is accessible to the numerate layperson. Can you improve it? Geometry guy 15:00, 13 May 2007 (UTC)[reply]

I, for one, think the level of the article is fine. The intro parts basically describe what each of the R.V represent, and the Memoryless property link seems to be fine. The main stumbling block for the real amateur is most likely the basic idea behind a "process". They can click on Stochastic Process and learn more about that.

I agree with Chassain(?, signed below) that the "process" is likely to be the stumbling concept for many readers. It makes sense that the level of articles varies in some ways among the stochastic, Markov, Bernoulli, Dirichlet, and Chinese restaurant process. At the same time the leads should provide a little more coherence and useful cross-reference. --P64 (talk) 21:26, 22 February 2010 (UTC)[reply]

application

On a unrelated note, are there practical uses for this simplistic process? If so, we should probably put that in. Akshayaj 21:21, 18 July 2007 (UTC)[reply]

Yes, if you consider mathematical applications practical. A number of important probabilistic models are based on Bernoulli processes, perhaps in disguise: For example, a one-dimensional simple random walk is defined in terms of a Bernoulli process, where each "coin flip" tells you whether to step left or right. Since any countable index set (e.g. the set of edges in a lattice) is equivalent to the integers, Bernoulli percolation is determined by a Bernoulli process in which 1s represent open edges and 0s represent closed edges. I'm sure there are other examples as well. 128.95.224.52 01:14, 19 October 2007 (UTC)[reply]

I would say, a huge number of applications, but, at the moment, I don't find many :-) It is historically important in data compression models (theoretical computer science), but Markovian sources or ergodic sources are considered more realistic. Besides percolation, there is also the Erdos Renyi model of random graphs, too.--Chassain (talk) 16:35, 3 January 2010 (UTC)[reply]

The article permits finite sequences of random variables(right?) to be processes. If that is wrong or unwise then it should be rewritten. So two coin tosses called first and second make a Bernoulli process. A single coin toss is a degenerate example.

Furthermore it is ubiquitous to formalize collections of observations as sequences. Given 20 observations "everyone" indexes them 1 to 20 although the sequence is purely formal; eg, there is no passage of time. So the Bernoulli process is the foundation of every binomial model (or application or whatever). --P64 (talk) 21:26, 22 February 2010 (UTC)[reply]

Bernoulli sequence

Is this standard usage, that the Bernoulli sequence is not a sequence of random variates zero and one but a sequence of index numbers (subset of N) where the random variate is one?

I have partly rewritten the article to fit this usage, which is a challenge, because it would be --and it has been for some other editors-- convenient to call the sequence of zero and one a Bernoulli sequence. This needs attention "urgently" inasmuch as I haven't finished rewriting for consistency.

Probably there is a big matter of pedagogical or encyclopedic tactics to resolve, which needs some group decision not simply one person's expertise in one subdiscipline. --P64 (talk) 23:11, 4 March 2010 (UTC)[reply]

I have added a statement in the main article to this effect. 24.1.53.152 (talk) 10:44, 29 January 2011 (UTC)[reply]

Bernoulli map

may be interpreted as the binary digital representation of a real number between zero and one.

First, that isn't unique. The rational numbers whose denominators are powers of 2, aka the multiples of powers of one-half, all have two binary digital representations. If we will gloss over that complication, it should at least be mentioned in a footnote. Many readers do know that 0.9999... = 1. This is a topological stumbling block. Does measure theory handle it without a glitch?

Second, this section needs to be rewritten after we decide the questions of categories, which may be a pedagogical or otherwise tactical rather than matters for deference to standard usage by experts in some subdiscipline.

When should we use the move that interprets a sequence as a single entity? (eg, a sequence of r.v., aka a discrete-time stochastic process, as a single random variable whose values are sequences of numbers)
When should we permit trials, and processes and their ilk (experiments, samples, etc) to live equally among the variates and also in the abstract space $\Omega$ ? (rather than restrict them as we restrict random variables to functions of Omega into R or Rn)