The bernoulli distribution is a discrete probability distribution, named after Swiss scientist Jacob Bernoulli, takes value '1' with probability 'p' and takes '0' with probability 'q'(1-p). The bernoulli trial has random outcome, can be success or can be failure.And for binomial distribution, if there are more then one event. And every event is independent. If we take success probability 'p' then failure probability will be '1-p'. And if we do 'n' trials with success probability 'p' then it is called binomial experiment, and it is denoted by B(n,p). Bernoulli distribution is a special form of binomial distribution & a member of exponential family.
If there are 'n' random indipendent variables with same success probability 'p', then binomial distribution will be:
and binomial(1,p) will be bernoulli distribution.
Probability mass function(PMF) is just a function which gives probability that a discrete random variable is equal to some value. This function gives probability of exactly k successes in the experiment B(n,p). Bernoulli experiment can lead us to find negative binomial geometric, many more other distribution.
Bernoulli probability mass function:
can also be like:
Binomial probability mass function:
Poisson probability mass function:
Poisson distribution is actually binomial distribution under the limiting condition that , & np = and is a constant. Under these restrictions, the limiting form of the Binomial distribution will be poisson distribution.
Some properties of network can be analyse through these distribution. poisson distribution shows it's use in fields of counting, for example vehicle movement on roads, accedents resord of city and where should we open new medicle centre for faster result.