Binomial Distribution
Introduction
The binomial distribution is a probability distribution used to model the number of successes in a given number of independent trials. The binomial distribution has been used to study a range of phenomena, from the probability of rolling a certain number of aces when rolling dice, to the probability of correctly predicting the outcome of a coin flip. The binomial distribution is a fundamental tool of statistical analysis and is used in many fields, including finance, economics, and medicine.
Definition
The binomial distribution is a probability distribution that models the number of successes (k) in a given number of independent trials (n) with a given probability of success (p). The binomial distribution is parameterized by two values: n (the number of trials) and p (the probability of success in each trial). The probability of observing k successes in n trials is given by the formula:
P(k; n, p) = (n!/(k! (n-k)!) ) (p^k) (1-p)^(n-k)
where n! denotes the factorial of n.
History
The binomial distribution was first studied by Blaise Pascal and Pierre de Fermat in the 17th century. They used the distribution to study the probability of achieving a certain number of successes in a given number of coin flips. The distribution was later studied by Jacob Bernoulli, who proved the general formula for calculating the probability of k successes in n trials, as described above.
Conclusion
The binomial distribution is an important tool for modeling the probability of achieving a certain number of successes in a given number of independent trials. The binomial distribution has been studied for centuries, and is used in many fields, including finance, economics, and medicine.
References
Pascal, B., & Fermat, P. (1654). Traitez de l’equilibre des lignes courbes. Paris: de l’Imprimerie Royale.
Bernoulli, J. (1713). Ars Conjectandi. Basel: Thurneysen.