An Introduction to Probability Theory and Its Applications, Volume 1

apply assume asymptotic Bernoulli trials binomial distribution cards central limit theorem Chapter 12 closed set coefficients coin conditional probability consider contains converges defined derived differential equations elements equal exactly example expected number finite fixed follows formula frequencies function gene genotype geometric distribution given Hence independent random variables inequality infinite integer interval large numbers law of large Markov chain Mathematical matrix means nth trial number of successes number of trials occurs P₁ particle player Pn(t Poisson distribution population positive possible Pr{A prob probability distribution problem proof prove random walk recurrent event replaced root run of length sample points sample space sequence of Bernoulli solution statistical statistically independent stochastic Suppose Table theory tion tossing transient transition probabilities trial number u₁ values Var(X variance X₁