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Important Random Variables Binomial: S X = { 0,1,2,..., n} Geometric: S X = { 0,1,2,... } Poisson: S X = { 0,1,2,... }

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Poisson Distribution: Used for modeling number of events in an interval or set if they occur randomly and independently. If N = Number of events in an interval T, Where α = Event rate T = Average number of events per size T Interval. e.g. Failure rate for chips in a system = 2 / year For a large number of Bernoulli trials with small success rate, p, Number of successes = Poisson

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P( k hits in n attempts) = with α = np as n → ∞ and np is constant. e.g.

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Examples of Poisson Distribution Applications 1) 2)

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Continuous Random Variables Uniform: Gaussian: Exponential:

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Mean: also called the expected value of X. If X is discrete E(X) does not exist for all random variables. It requires that:

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Variance: Standard Deviation = √Var Variance measures the dispersion of X about the mean. Moments : nth Moment (X) = nth Central Moment = nth Absolute Moment = nth Generalized Moment about a =

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Markov Inequality Proof:

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Chebychev Inequality: Beinayme Inequality: The Chebychev inequality is a special case of this with b=m, n=2

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Memoryless Random Variables: A random variable, X, is called memoryless if, for h>0 i.e. the incremental probability of x+h is independent of x. Context is meaningless. Geometric is the only memoryless discrete random variable. Exponential is the only memoryless continuous random variable.

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Gaussian Random Variable The Gaussian distribution is also called the normal distribution and is often popularly referred to as the bell curve. It is found to be a good model for random variables in many real-world systems, and has many useful properties (as we will see later in the class). m f X (x) There is no closed form for F X (x) x

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Standard Gaussian Random Variable Note: The textbook uses instead of G, but we will later use for something else.

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De Moivre-Laplace Theorem: if np(1-p) >> 1 where m = np and = np(1-p)

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Functions of Random Variables: If X is a random variable Y = g(X) is also a random variable. Any event {g(X) ≤ a} can be seen as a union of events in S X. This is called the equivalent event. e.g. g(x)g(x) a i j k x

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In general, if y=g(X) has n solutions, {x k } y+dy x 1 + dx 1 y x1x1 x2x2 x 2 + dx 2 x 3 + dx 3 x3x3 Example:

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Linear Case: Since Y is linear in X, there is only one solution to y=ax + b : and

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Chapter 6 Continuous Random Variables and Probability Distributions

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