Topics: Stochastic Process- Bernoulli Trial


A stochastic process is called a Bernoulli process if it satisfies the following conditions:

  1. are independent
  2. ,

We’ll refer to the event as success and to as failure.

Number of Successes


Let be a Bernoulli process with a success probability of .

The number of successes we’ve had up to the th trial is denoted by and defined as:

Note that defines another stochastic process, one with discrete state and parameter sets.


From the previous definition, we can tell that is the number of successes we’ve had along the trials .

Distribution of and


Each of these individual follow a Bernoulli distribution with a parameter .

As such, follows a binomial distribution with parameters , since the sum of independent Bernoulli-distributed random variables distributes binomially.


Now, we can tell that for :

Basically what was already established for a Bernoulli distribution.