6.262.Lec2

# 6.262.Lec2 - DISCRETE STOCHASTIC PROCESSES Lecture 2...

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Lecture 2 - 2/5/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 2 Review: Random variables, (Cumulative) Distribution functions Expectations Functions of random variables Alternate method to calculate expectations. Sums of IID random variables. A strong form of convergence: Convergence in mean square Review: Central limit theorem. A weak form of convergence: Convergence in distribution Review: Markov and Chebyshev inequalities Moment Generating Functions and the Chernoff Bound

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Lecture 2 - 2/5/2010 Discrete Stochastic Processes 2 RANDOM VARIABLES A random variable is neither random nor a variable. (Discuss amongst yourselves.) A random variable X in a probability model is a function from the sample space to the (finite) real numbers. P X (x) = P(X= x) = for countable sample space Ω ; a Ω such that X(a) = x Px X a f called the Probability Mass Function of X . In general, PX x P aXa x ≤= af a f lq ch : . Fx PX x X a f =≤ is called the Distribution Function of X ; for given r.v. X , it is a function of a real variable x . If it exists, fx d Fxd x XX = / is called the density of X and X is called a continuous random variable . If X takes on only a countable set of values, it is called discrete . () Pa
Lecture 2 - 2/5/2010 Discrete Stochastic Processes 3 The (Cumulative) Distribution Function 1 F (x) x 0 x Fx X af is monotonic non-decreasing and always exists for all random variables. What does the height and placement of a jump tell you? The density function and the PMF are usually more convenient for hand calculations and thus more familiar in elementary subjects; the distribution function is more useful conceptually since discrete, continuous, and arbitrary random variables are all described by probability distribution functions. If XY Z ,,, . . . are random variables in a probability model with sample space S , then the Joint Distribution Function is: () ( ) ( ) ( ) ( ) { } ( ) z a Z y a Y x a X a P z Z y Y x X P z y x F XYZ Ω = = , , : , , , ,

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Lecture 2 - 2/5/2010 Discrete Stochastic Processes 4 The joint density (if it exists) is f x y z d F x y z dxdydz ,, ,, / a f a f = 3 . X , Y , Z are (statistically) independent if Fx y z F x F y F z XYZ X Y Z a f a f a f a f = for all x , y , z . Pairwise independence does not imply independence:
Lecture 2 - 2/5/2010 Discrete Stochastic Processes 5 EXPECTATIONS The expectation of X is EX xP x X x = a f for a discrete random variable X x f xd x X = −∞ z af for continuous variables x dF x X = −∞ z a f in general This is a Stieltjes integral and can be regarded as shorthand for any sensible way of integrating (view dF x X as fx d x X with the density function including impulses, etc).

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## This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.Lec2 - DISCRETE STOCHASTIC PROCESSES Lecture 2...

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