Introduction to Probability and Stochastic Processes with Applications

INTRODUCTION

Since its origin, probability theory has been linked to games of chance. In fact by the time of the first roman emperor, Augustus (63 B.C.–14 A.D.), random games were fairly common and mortality tables were being made. This was the origin of probability and statistics. Later on, these two disciplines started drifting apart due to their different objectives but always remained closely connected. In the sixteenth century philosophical discussions around probability were held and Italian philosopher Gerolamo Cardano (1501–1576) was among the first to make a mathematical approach to randomness. In the seventeenth and eighteenth centuries major advances in probability theory were made due in part to the development of infinitesimal calculus; some outstanding results from this period include: the law of large numbers due to James Bernoulli (1654–1705), a basic limit theorem in modern probability which can be stated as follows: if a random experiment with only two possible outcomes (success or failure) is carried out, then, as the number of trials increases the success ratio tends to a number between 0 and 1 (the success probability); and the DeMoivre-Laplace theorem (1733, 1785 and 1812), which established that for large values of n a binomial random variable with parameters n and p has approximately the same distribution of a normal random variable with mean np and variance np(1 − p). This result was proved by DeMoivre in 1733 for the case p = and then extended to arbitrary ...