Editing Bernoulli process (section)

=== Interpretation ===

The two possible values of each ''X''<sub>''i''</sub> 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''<sub>''i''</sub> may be called [[Bernoulli trial]]s with parameter p.

In many applications time passes between trials, as the index i increases. In effect, the trials ''X''<sub>1</sub>,&nbsp;''X''<sub>2</sub>,&nbsp;...&nbsp;''X''<sub>i</sub>,&nbsp;... happen at "points in time" 1,&nbsp;2,&nbsp;...,&nbsp;''i'',&nbsp;.... That passage of time and the associated notions of "past" and "future" are not necessary, however. Most generally, any ''X''<sub>i</sub> and ''X''<sub>''j''</sub> in the process are simply two from a set of random variables indexed by {1,&nbsp;2,&nbsp;...,&nbsp;''n''}, the finite cases, or by {1,&nbsp;2,&nbsp;3,&nbsp;...}, the infinite cases. 

One experiment with only two possible outcomes, often referred to as "success" and "failure", usually encoded as 1 and 0, can be modeled as a [[Bernoulli distribution]].<ref name=":0">{{Cite book|title=A modern introduction to probability and statistics|isbn=9781852338961|pages=45–46|last1=Dekking|first1=F. M.|last2=Kraaikamp|first2=C.|last3=Lopuhaä|first3=H. P.|last4=Meester|first4=L. E.|year=2005|publisher=Springer }}</ref> Several random variables and [[probability distribution]]s beside the Bernoullis may be derived from the Bernoulli process:

*The number of successes in the first ''n'' trials, which has a [[binomial distribution]] B(''n'',&nbsp;''p'')
*The number of failures needed to get ''r'' successes, which has a [[negative binomial distribution]] NB(''r'',&nbsp;''p'')
*The number of failures needed to get one success, which has a [[geometric distribution]] NB(1,&nbsp;''p''), a special case of the negative binomial distribution

The negative binomial variables may be interpreted as random [[Negative binomial distribution#Waiting time in a Bernoulli process|waiting times]].