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In probability theory and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables $ X_i $ are identically distributed 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.

Definition

A Bernoulli process is a finite or infinite sequence of independent random variables $ X_1, X_2, X_3, \ldots, $ 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.

Interpretation

The two possible values of each $ X_i $ are often called "success" and "failure". Thus, when expressed as a number 0 or 1, the outcome may be called the number of successes on the $ i $th "trial".

Two other common interpretations of the values are true or false and yes or no. Under any interpretation of the two values, the individual variables $ X_i $ may be called Bernoulli trials with parameter $ p $.

In many applications time passes between trials, as the index i increases. In effect, the trials $ X_1, X_2, \ldots, X_i, \ldots $ happen at "points in time" $ 1, 2, \ldots, i, \ldots $. That passage of time and the associated notions of "past" and "future" are not necessary, however. Most generally, any $ X_i $ and $ X_j $ in the process are simply two from a set of random variables indexed by $ \{1, 2, \ldots, n\} $ or by $ \{1, 2, 3, \ldots\} $, the finite and infinite cases.

Several random variables and probability distributions beside the Bernoullis may be derived from the Bernoulli process:

Formal definition

Using the example of the Tech card in a A8Box Daily Kit Box Daily Kit Box † which can be either Advanced Tech $ \{A\} $ or Mid-Tech $ \{M\} $, the Bernoulli process can be formalized in the language of probability spaces as a random sequence of independent realisations of a random variable that can take values of Advanced or Mid-Tech. The state space for an individual value is denoted by $ 2=\{A,M\} $.

Specifically, one considers the countably infinite direct product of copies of $ 2=\{A,M\} $. It is common to examine either the one-sided set $ \Omega=2^\mathbb{N}=\{A,M\}^\mathbb{N} $ or the two-sided set $ \Omega=2^\mathbb{Z} $.

If the chances of revealing Advanced Tech or Mid-Tech are given by the probabilities $ \{p,1-p\} $, then one can define a natural measure on the product space, given by $ P=\{p, 1-p\}^\mathbb{N} $ (or by $ P=\{p, 1-p\}^\mathbb{Z} $ for the two-sided process). Given a specific sequence of revealed cards $ [\omega_1, \omega_2,\ldots\omega_n] $ at times $ 1,2,\ldots,n $, the probability of observing this particular sequence is given by

$ P([\omega_1, \omega_2,\ldots ,\omega_n])= p^k (1-p)^{n-k} $

where $ k $ is the number of times that $ A $ appears in the sequence, and $ n-k $ is the number of times that $ M $ appears in the sequence. There are several different kinds of notations for the above; a common one is to write

$ P(X_1=x_1, X_2=x_2,\ldots, X_n=x_n)= p^k (1-p)^{n-k} $

where each $ X_i $ is a binary-valued random variable with

  • $ x_i=1 $ if $ \omega_i=A $ or
  • $ x_i=0 $ if $ \omega_i=M $.

Note that the probability of any specific, infinitely long sequence of revealed cards is exactly zero; this is because

$ \lim_{n\to\infty}p^n=0 $ for any $ 0\le p<1 $.

Law of large numbers

Main article: Law of large numbers

Let us assume the canonical process with $ A $ represented by $ 1 $ and $ M $ represented by $ 0 $. The law of large numbers states that, on the average of the sequence, i. e., $ \bar{X}_{n}:=\frac{1}{n}\sum_{i=1}^{n}X_{i} $, will approach the expected value almost surely, that is, the events which do not satisfy this limit have zero probability. The expected value of revealing Advanced Tech, assumed to be represented by 1, is given by $ p $. In fact, one has

$ \operatorname{E}[X_i]=P([X_i=1])=p $,

for any given random variable $ X_i $ out of the infinite sequence of Bernoulli trials that compose the Bernoulli process.

Binomial distribution

Main article: Binomial distribution

One is often interested in knowing how often one will observe $ A $ in a sequence of $ n $ revealed cards. This is given by simply counting: Given $ n $ successive revealed cards, that is, given the set of all possible strings of length $ n $, the number $ N(k,n) $ of such strings that contain $ k $ occurrences of $ A $ is given by the binomial coefficient

$ N(k,n) = {n \choose k}=\frac{n!}{k! (n-k)!} $

If the probability of revealing Advanced Tech is given by $ p $, then the total probability of seeing a string of length $ n $ with $ k $ Advanced Tech cards is

$ P([S_n=k]) = {n\choose k} p^k (1-p)^{n-k} $, where $ S_n=\displaystyle \sum_{i=1}^{n}X_i $.

The probability measure thus defined is known as the binomial distribution.

See also

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