Binomial Stochastic Process

A Binomial Stochastic Process, [math]\displaystyle{ B(n,p) }[/math], is a discrete-time discrete-outcome stochastic process composed of [math]\displaystyle{ n }[/math] mutually independent binomial trials with [math]\displaystyle{ p }[/math] probability of success.

• Context:
  • Output, [math]\displaystyle{ s }[/math] successes, where [math]\displaystyle{ 0\le s \le n }[/math].
  • The posterior distribution for a binomial process parameter [math]\displaystyle{ p }[/math] takes the shape of a beta distribution [math]\displaystyle{ Beta(x;α,β) }[/math].
• Example(s):
  • a Gambling Slot Machine.
  • a Lottery Game.
  • …
• Counter-Example(s):
  • an Non-Binomial Multinomial Process.
  • a Stochastic Binary Variable.
• See: Binomial Probability Function, Binomial Regression, Compound Random Experiment.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Bernoulli_process Retrieved:2015-6-20.
  • In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables X_i are identical and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable X_i in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme.

    The problem of determining the process, given only a limited sample of the Bernoulli trials, may be called the problem of checking whether a coin is fair.

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Bernoulli_process#Definition Retrieved:2015-6-20.
  • A Bernoulli process is a finite or infinite sequence of independent random variables X₁, X₂, X₃, ..., such that
    • For each i, the value of X_i is either 0 or 1;
    • For all values of i, the probability that X_i = 1 is the same number p.
  • In other words, a Bernoulli process is a sequence of independent identically distributed Bernoulli trials.

    Independence of the trials implies that the process is memoryless. Given that the probability p is known, past outcomes provide no information about future outcomes. (If p is unknown, however, the past informs about the future indirectly, through inferences about p.)

    If the process is infinite, then from any point the future trials constitute a Bernoulli process identical to the whole process, the fresh-start property.

2011

• http://www.vosesoftware.com/ModelRiskHelp/index.htm#Probability_theory_and_statistics/Stochastic_processes/The_binomial_process.htm
  • QUOTE: A binomial process is a random counting system where there are n independent identical trials, each one of which has the same probability of success p, which produces s successes from those n trials (where 0 ≤ s ≤ n and n > 0 obviously). There are thus three parameters {n, p, s} that between them completely describe a binomial process. Associated with each of these three parameters are three distributions that describe the uncertainty about or variability of these parameters. The three distributions require that one has knowledge of two parameters in order to use these distributions to estimate the third.

2009

• http://www.teacherlink.org/content/math/interactive/probability/glossary/glossary.html
  • Bernoulli Trial: Another name for a trial in a binomial experiment.

2006

• (Dubnicka, 2006f) ⇒ Suzanne R. Dubnicka. (2006). “Special Discrete Distributions - Handout 6." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • BERNOULLI TRIALS: Many experiments consist of a sequence of trials, where
    • (i) each trial results in a “success” or a “failure,”
    • (ii) there are n trials (where n is fixed),
    • (iii) the trials are independent, and
    • (iv) the probability of “success,” denoted by p, 0 < p < 1, is the same on every trial.
  • TERMINOLOGY : In a sequence of n Bernoulli trials, denote by X the number of successes (out of n). We call X a binomial random variable, and say that “X has a binomial distribution with parameters n and success probability p.” Shorthand notation is X ~ B(n, p).

2005

• (Lord et al., 2005) ⇒ Dominique Lord, Simon P. Washington, and John N. Ivan. (2005). “Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory.” In: Accident Analysis & Prevention, 37(1). doi:10.1016/j.aap.2004.02.004
  • QUOTE: … where [math]\displaystyle{ p }[/math] is the probability of success (a crash) and [math]\displaystyle{ q = (1-p) }[/math] is the probability of failure (no crash).
    In general, if there are [math]\displaystyle{ N }[/math] independent trials (vehicles passing through an intersection, road segment, etc.) that give rise to a Bernoulli distribution, then it is natural to consider the random variable [math]\displaystyle{ Z }[/math] that records the number of successes out of the N trials. Under the assumption that all trials are characterized by the same failure process (this assumption is revisited later in the paper), the appropriate probability model that accounts for a series of Bernoulli trials is known as the binomial distribution, and is given as: :[math]\displaystyle{ P(Z=n) = {{N} \choose {n}} p^n(1-p)^{N-n} \tag{1} }[/math] where [math]\displaystyle{ n = 0,1,2,...,N }[/math]. In equation (1), [math]\displaystyle{ n }[/math] is defined as the number of crashes or collisions (successes). The mean and variance of the binomial distribution are [math]\displaystyle{ E(Z) = Np }[/math] and [math]\displaystyle{ VAR(Z) = Np(1-p) }[/math] respectively.

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