6.262.Lec1

6.262.Lec1 - DISCRETE STOCHASTIC PROCESSES Probability...

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Lecture 1 - 2/3/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 1 Probability Discrete stochastic processes What this course is about Some application areas. The random processes we will study: Counting Processes Renewal Processes Poisson Processes Queues Markov Processes Finite state, discrete-time Markov chains Countable state, discrete-time Markov chains. Countable state, continuous-time Markov Processes Martingales
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Lecture 1 - 2/3/2010 Discrete Stochastic Processes 2 Probability A Stochastic Process is a special case of a probability model/experiment Probability Model (Quick, very basic review): Sample space, (set of possible "elementary outcomes" ) Events (subsets of sample space) Probability associated with each event, "Experiment" results in one & only one sample point Probability of the union of mutually exclusive events (disjoint sets) is the sum of the probabilities of the events. Σ : P R Σ ⎯⎯→ Ω ω ∈Ω
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Lecture 1 - 2/3/2010 Discrete Stochastic Processes 3 D iscrete Probability Model—Finite or Countable Sample Space In general, for finite or countable sample space, each sample point a i has probability Pa i af and for each event E , PE i aE i = .
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Lecture 1 - 2/3/2010 Discrete Stochastic Processes 4 Continuous Probability Models—Uncountable Sample Space Assigning probabilities to sample points doesn't always work: Ex. 1 Uniform distribution over real numbers 0 to 1 – each number has 0 probability; Ex. 2 Unending binary sequence X 1 , X 2 ,... where each bit is 1 with probability p ; bits are statistically independent -- each sequence has 0 probability . We must work with events (intervals, unions of intervals, . ..) directly. Requirements on limits of series of events is what makes formal probability theory (measure theory) tricky. Possible events for Ex. 2. Event that first hundred bits are all 0. Event that lim ... n n X X X n p →∞ + + + = 12 (all sequences of {0,1} that satisfy equation)
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Lecture 1 - 2/3/2010 Discrete Stochastic Processes 5 Stochastic Processes and Discrete Stochastic Processes A stochastic process is a probability model in which each sample point (i.e. elementary outcome) is actually a function of "time". The above binary sequence is a stochastic process over discrete (integer) time. For any probabilistic model representing the times at which customers or thigamajigs arrive, one can take the number of arrivals up to t (as a function of t ) as a sample function. The probabilistic model must in principle specify the probability of each event (e.g. having 57 arrivals between time 10 and 100 is an event). A discrete stochastic process is one in which the sample functions change by discrete amounts (discontinuously) as functions of time. (You may be used to Gaussian process where functions are continuous in time.) In discrete stochastic processes, we concentrate on when the function changes.
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Lecture 1 - 2/3/2010 Discrete Stochastic Processes 6 Models It is important in all fields to separate models from reality. In probability, it is particularly important, since only one sample point is " real " .
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6.262.Lec1 - DISCRETE STOCHASTIC PROCESSES Probability...

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