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1 Introduction to Stochastic Processes

MA636: Introduction to Stochastic processes1 11 Introduction to Stochastic IntroductionStochastic modelling is an interesting and challenging area of proba-bility and statistics. Our aims in this introductory section of the notesare to explain what astochastic processis and what is meant by theMarkov property, give examples and discuss some of the objectivesthat we might have in studying Stochastic DefinitionsWe begin with a formal definition, Astochastic processis a familyof random variables{X }, indexed by a parameter , where belongsto some index set .In almost all of the examples that we shall look at in this module, will represent time. If is a set of integers, representing specifictime points, we have a Stochastic process indiscrete timeand weshall replace the general subscript byn.

MA636: Introduction to stochastic processes 1–3 examples of all four combinations (discrete/continuous time in con-junction with discrete/continuous random variable) in this module. We end this section with a few more definitions related to stochastic processes: • A counting process is a process X(t) in discrete or continuous

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