Theory 1 - Bernoulli, binomial, geometric, Pascal, uniform
In a Bernoulli process, an experiment with binary outcomes is repeated; for example flipping a coin repeatedly. Several discrete random variables may be defined in the context of some Bernoulli process.
Notice that the sample space of a Bernoulli process is infinite: an outcome is any sequence of trial outcomes, e.g.
Bernoulli variable
A random variable
is a Bernoulli indicator, written , when indicates whether a success event, having probability , took place in trial number of a Bernoulli process. Bernoulli indicator PMF:
An RV that always gives either
Binomial variable
A random variable
is binomial, written , when counts the number of successes in a Bernoulli process, each having probability , over a specified number of trials. Binomial PMF:
- For example, if
, then gives the odds that success happens exactly 5 times over 10 trials, with probability of success for each trial. - In terms of the Bernoulli indicators, we have:
- If
is the success event, then , and is the probability of failure.
Geometric variable
A random variable
is geometric, written , when counts the discrete wait time in a Bernoulli process until the first success takes place, given that success has probability in each trial. Geometric PMF:
- For example, if
, then gives the probability of getting: failure on the first trials AND success on the trial.
Pascal variable
A random variable
is Pascal, written , when counts the discrete wait time in a Bernoulli process until success happens times, given that success has probability in each trial. Pascal PMF:
- For example, if
, then gives the probability of getting: the success on (precisely) the trial. - Interpret the formula:
ways to arrange successes among ‘prior’ trials, times the probability of exactly successes and failures in one particular sequence. - The Pascal distribution is also called the negative binomial distribution, e.g.
.
Uniform variable
A discrete random variable
is uniform on a finite set , written , when the probability is a fixed constant for outcomes in and zero for outcomes outside . Discrete uniform PMF:
Continuous uniform PDF: