Bernoulli Distribution Fitting
In the realm of probability theory and statistics, the Bernoulli distribution stands as a foundational concept, named in honor of the Swiss mathematician Jacob Bernoulli. This distribution characterizes the discrete probability of a random variable, which assumes the value of 1 with a likelihood denoted as p, and the value of 0 with the complementary probability, q = 1  p.
To understand it more intuitively, envision a scenario where a single experiment poses a binary question, eliciting a yes or no response. The Bernoulli distribution provides a framework for modeling such outcomes. These yesno inquiries yield results that can be represented by a Boolean variable: a single bit of information that holds a value of success, yes, true, or one with a probability of p, and failure, no, false, or zero with a probability of q.
One common illustration of this distribution lies in its application to coin tosses. Imagine tossing a coin?biased or unbiased?where the outcomes are conventionally labeled as "heads" and "tails." Here, the Bernoulli distribution offers a means to quantify the probabilities associated with these outcomes. If p represents the probability of the coin landing on heads, then 1 would signify a head, while 0 would signify a tail. In cases of biased coins, where the probabilities of heads and tails are unequal, p deviates from the standard 1/2.
In summary, the Bernoulli distribution serves as a fundamental tool in modeling binary events, encapsulating the essence of probability in scenarios where outcomes are dichotomous. Whether applied to coin flips or other yesno inquiries, its simplicity and versatility render it indispensable in various statistical analyses and decisionmaking contexts.
The binomial distribution is the discrete probability distribution of the number of
successes in a sequence of >n independent yes/no experiments, each of which
yields success with probability p. Such a success/failure experiment is also
called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial
distribution is a Bernoulli distribution.
References:
Where:

p is the success probability for each trial

q is the failure probability for each trial

f(k,n,p) is probability of k successes in n trials when the success probability is p
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