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Unformatted text preview: Random Variables and Probability Distributions Enrique CamposN anez The George Washington University ApSc 116 Table of Contents Random Variables Common Discrete Distributions Continuous Random Variables and Distributions The Distribution Function Random Variable Definition (Random Variable) A random variable (r.v.) is a realvalued function defined on a sample space S , i.e., a r.v. X is a function that assigns a real number X ( s ) to each possible outcome s S . Notation: Random variables will be given capital letter ( X ), while their specific values will be written using lower case letters ( x ). Example (Tossing a Coin) The experiment consists of tossing a coin 10 times. The sample space is of size 2 10 . A random variable X can be defined as the number of heads in the 10 tosses. Therefore for the sequence s = { HHTTTHHTTH } , will be X ( s ) = X ( HHTTTHHTTH ) = 5 . The Distribution of a Random Variable Given a probability distribution (or measure) specified on the sample space S , we can determine probabilities for the different possible values of a r.v. Let A be any subset of the real numbers; then Pr ( X A ) denotes the probability that the outcome s of the experiment will be such that X ( s ) A . That is, Pr ( X A ) = Pr ( { s : X ( s ) A } ) . Example (Tossing a coin) In the same example described in the previous slide (10 coin tosses), one can compute the probability Pr ( X = x ) = 10 x 1 2 10 , for x = 0 , 1 , 2 ,..., 10. Discrete Distribution Definition We say a r.v. has a discrete distribution, or that X is a discrete r.v., if X can take only a finite number of different values x 1 ,..., x k or, at most, an infinite, but numerable set of values x 1 , x 2 ,......
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 Fall '08
 TomMazzuchi

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