Cadeias de Markov e Martingais: uma aplicação nas urnas de Pólya

Abstract

In this work we introduce two subjects of great importance the Probability theory: i) Markov Chains and ii) Martingales and in the end, we ilustrate both subjects with the Pólya urn model. Weve shown the Theorem of existence and uniqueness of stationary distribution, theMartingale convergency theorem and some results of the Pólya urn model with the inicial configuration of W0 >= 1 white balls and B0 >= 1 black balls, and returns of a >= 1 additional balls of the same collor of that one drawn. Weve seen that i) the ammount of black balls in the k-th draw follows a Betha (or Uniform) distribution and, ii) the probability of drawing a black ball in at any instant k follows a Bernoulli distribution.