An Introduction to Probability Theory and Its Applications, Volume 1

apply approximation assume asymptotic Bernoulli trials binomial distribution cards central limit theorem Chapter 12 closed set coefficients coin conditional probability consider contains converges corresponding defined derived differential equations elements equal exactly example Find the probability finite fixed follows formula function gambler genotype geometric distribution given Hence independent random variables infinite initial integer interval large numbers law of large Markov chain Markov process Mathematical matrix mean recurrence method nth trial number of trials occurs P₁ particle Pik(t player Pn(t Poisson distribution population positive possible Pr{A Pr{X prob probability distribution problem proof prove random walk recurrent event replaced roots run of length s₁ sample points sample space solution statistical stochastic Suppose theory tion tossing transient transition probabilities trial number values Var(X variance X₁