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Week4-Class3-Random Variables

Week4-Class3-Random Variables - Random Variables and...

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Unformatted text preview: Random Variables and Probability Distributions Enrique Campos-N´ a˜nez 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 real-valued 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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Week4-Class3-Random Variables - Random Variables and...

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