Multinomial Distribution
The multinomial distribution is a fundamental probability distribution used to model events with multiple outcomes. It is used to understand the probability of discrete random variables taking on multiple values. This distribution is used in a variety of applications, from medical research to engineering. The multinomial distribution is a generalization of the binomial distribution, which applies to two outcomes. This article will discuss the properties and applications of the multinomial distribution.
Definition
The multinomial distribution can be thought of as a generalization of the binomial distribution. It is used to describe the probability of discrete random variables taking on multiple values. The probability of each value is described by a vector, p=(p_1,…,p_n)^T, where the sum of the elements is 1 (∑_i p_i=1). The probability of an event can then be calculated as
P(x_1=k_1,…,x_n=k_n)=frac{n!}{k_1!…k_n!}p_1^{k_1}…p_n^{k_n},
where n is the number of trials, k_i is the number of times the i^{th} outcome occurs, and p_i is the probability of the i^{th} outcome.
Properties
The multinomial distribution has several important properties. For example, it is a discrete probability distribution, meaning that it contains no continuous values. Additionally, the multinomial distribution is a symmetric distribution, meaning that the probability of each outcome is the same. Finally, the distribution is also a finite distribution, meaning that the probability of any single outcome is finite.
Applications
The multinomial distribution is used in a variety of applications. For example, it is used in medical research to model the probability of different outcomes, such as the probability of a patient having a certain disease. It is also used in engineering to model the probability of different failures in a system. Finally, the multinomial distribution is also used in finance to model the probability of different returns from investments.
Conclusion
The multinomial distribution is a fundamental probability distribution used to model events with multiple outcomes. It is used to understand the probability of discrete random variables taking on multiple values. This distribution is used in a variety of applications, from medical research to engineering. The multinomial distribution is a generalization of the binomial distribution, which applies to two outcomes. This article discussed the properties and applications of the multinomial distribution.
References
Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2013). Bayesian data analysis (3rd ed.). Boca Raton, FL: Chapman & Hall/CRC.
Gourieroux, C., & Monfort, A. (1996). Statistics and financial applications of the multinomial distribution. Journal of Econometrics, 72(1), 49-68.
Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), 97-109.