Binomial Stochastic Process

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

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).