Write short notes on Bernoulli distribution. Also write the use of Bernoulli process

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability p and value 0 with failure probability q = 1 − p. So if X is a random variable with this distribution, we have:

Definition:

A Bernoulli process is a discrete-time stochastic process consisting of a finite or infinite sequence of independent random variables X₁, X₂, X₃,..., 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. The two possible values of each X_i are often called "success" and "failure", so that, when expressed as a number, 0 or 1, the value is said to be the number of successes on the ith "trial". The individual success/failure variables X_i are also called Bernoulli trials.

Independence of Bernoulli trials implies memory lessness property: past trials do not provide any information regarding future outcomes. From any given time, future trials are also a Bernoulli process independent of the past (fresh-start property).

Random variables associated with the Bernoulli process include:

• The number of successes in the first n trials; this has a binomial distribution;
• The number of trials needed to get r successes; this has a negative binomial distribution.
• The number of trials needed to get one success; this has a geometric distribution, which is a special case of the negative binomial distribution.

The problem of determining the process, given only a limited sample of Bernoulli trials, is known as the problem of checking if a coin is fair.

Bernoulli distribution:

One widely used probability distribution of a discrete random variable is the binomial distribution. It describes a variety of processes of interest to managers. The binomial distribution describes discrete, not continuous, data, resulting from an experiment known as a Bernoulli process, after the seventeenth-century Swiss mathematician Jacob Bernoulli. The tossing of a fair coin a fixed number of times is a Bernoulli process, and the outcomes of such tosses can be represented by the binomial probability distribution. The success or failure of interviewees on an aptitude test may also be described by a Bernoulli process. On the other hand, the frequency distribution of the lives of fluorescent lights in a factory would be measured on a continuous scale of hours and would not quality as a binomial distribution.

Use of the Bernoulli process:

We can use the outcomes of a fixed number of tosses of a fair coin as an example of a Bernoulli process. We can describe this process as follows:

1. Each trial (each toss, in this case) has only two possible outcomes: heads or tails, yes or no, success or failure.

2. The probability of the outcome of any trial (toss) remains fixed over time with a fair coin, the probability of heads remains 0.5 for each toss regardless of the number of times the coin is tossed.

3. The trials are statistically independent; that is, the outcome of one toss does not affect the outcome of any other toss.