Bernoulli distribution is distribution where two possible outcome exists, probability of success “p” and probability of failure “q=1-p”. This outcome is known as Bernoulli trial.
Most of the discrete distribution are related with Bernoulli trials. An experiment is called Bernoulli trial if it has two possible outcomes namely success and failure. The probability of success as well as the probability of failure of a Bernoulli trial remains the same from trial to trial. If p is the probability of success then q=1-p is the probability of failure. The name of Bernoulli trial is given in honor of James Bernoulli, who wrote in Latin, about the year of 1700 an important work on probability titled “Ars Conjectandi”.
A discrete random variable X is said to have a Bernoulli distribution if its probability function is defined as,
![\[ f(x;p)=p^{x}q^{1-x}; for..x=0,1 \]](https://www.statisticalaid.com/wp-content/ql-cache/quicklatex.com-e2de7597880adaa1a31ff8f38adb6d50_l3.png "Rendered by QuickLaTeX.com")
Here , p is the parameter of the distribution which satisfying 0<=p<=1 and p+q=1.
Mean= E[X]= p and,
Variance=Var[X]= pq
If q<1 then the mean of the Bernoulli distribution is greater than the variance. that means,
E[X]>Var[X] ; when q<1.
Properties
- An experiment with two possible outcomes regarding as success and failure, the probability of success and failure is always remain same and the probability is 1/2 (half).
- Mean of Bernoulli Distribution is p.
- Variance of Bernoulli Distribution is pq.
- Moment generating function is .
.Applications of Bernoulli distribution
Bernoulli distribution is a form of Binomial distribution where N=1. If N>1 then it is converted into Binomial distribution. so the real life application of Bernoulli distribution is very rare because N is greater than 1 almost all the study. we consider the applications as following,
- There are a lot of areas where the application of binomial theorem is inevitable, even in the modern world areas such as computing. In computing areas, binomial theorem has been very useful such a in distribution of IP addresses. With binomial theorem, the automatic distribution of IP addresses is not only possible but also the distribution of virtual IP addresses.
- Another field that used Binomial Theorem as the important tools is the nation’s economic prediction. Economists used binomial theorem to count probabilities that depend on numerous and very distributed variables to predict the way the economy will behave in the next few years. To be able to come up with realistic predictions, binomial theorem is used in this field.
- Binomial Theorem has also been a great use in the architecture industry in design of infrastructure. It allows engineers, to calculate the magnitudes of the projects and thus delivering accurate estimates of not only the costs but also time required to construct them. For contractors, it is a very important tool to help ensuring the costing projects is competent enough to deliver profits.