Binomial Stochastic Process

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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.



References

2015


  • (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 X1X2X3, ..., such that
      • For each i, the value of Xi is either 0 or 1;
      • For all values of i, the probability that Xi = 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

2009

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