Bernoulli distribution

The Bernoulli distribution is a theoretical model used to represent a discrete random variable which can only result in two mutually exclusive events. 

In other words, Bernoulli’s distribution is a distribution applied to a discrete random variable, which can only result in two possible events: “success” and “no success”.

Recommended articles: sample space , Bernoulli distribution example and Laplace Rule.

Bernoulli experiments

An experiment is a random action which we have no way of predicting, such as the result of throwing a dice. In the Bernoulli distribution we only do a single experiment . In the case that more than one experiment is performed, as in the Binomial distribution , the experiments are independent of each other.

“Success” and “and not success”

They are experiments where the final situation can only result in two exclusive results or events:

  • The result we expect to happen. That is, ” success.”
  • The result other than the result we expect to occur. That is, ” no success.”

P parameter

Given a discrete random variable Z whose frequency can satisfactorily approximate a Bernoulli distribution with a parameter p.

The frequency of the random variable Z can be approximated satisfactorily by a Bernoulli distribution with probability p.

Generally the parameter p is used to indicate the probability of success of the discrete random variable Z. Then:

Possible results of the random variable Z.

  • If the random variable Z results in the result that we had defined as “success” at the beginning of the experiment, (Z = 1), then the probability of obtaining that particular result is (p).
  • If the variable Z results in a different result than the one we had defined as “not successful” at the beginning of the experiment, (Z = 0), then the probability of obtaining that specific result is (1-p).


It is important to note that the result ” not successful ” does not refer to the opposite of “success”, but refers to any case other than the one that represents “success” as long as there are more than two possibilities.

That is, in the case of rolling a dice, if the variable “success” refers to obtaining a four (4) in a roll, the variable “not success” will be any result other than four (4) that we can obtain in a roll. lying

Sample space : {1,2,3,4,5,6}.

In the case of a coin (not tricked), we can only get two possible results: face or cross. Then, in this case the variable “no success” will be effectively the opposite of the variable “success”.

Sample space: {1,2}.

Parameter formula p and the Laplace Rule:

To obtain the p parameter we use the Laplace Rule:

Laplace rule.

  • Possible cases:These are all possible results that we can obtain in an experiment. For example, if the experiment is to roll a dice, we will have six (6) possible cases because a dice has only six (6) faces.
  • Probable cases: These are the results that appear in each experiment sequentially , that is, the results are exclusive : if one result occurs, the others cannot occur. In the experiment of throwing a dice, each face of the dice is a probable case. In other words, a two (2) or a five (5) result are examples of probable cases in the experiment of throwing a dice.

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