handout_week2_a - following two conditions but is otherwise...

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Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Definition of a Random Variable Many random phenomena have outcomes that are sets of real numbers: arrival time of the next customer at a movie theater, . .. Even when a sample space S is not numerical, we might want to generate a new sample space from S that is numerical. To obtain numerical outcomes, we need a rule or mapping from the original sample space to the real line. The function or rule that assigns values to each outcome is fixed and deterministic. Since the outcomes are random, the results of this mapping will also be random.
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Definition of a Random Variable A random variable is a number X ( ξ ) assigned to every outcome ξ in the sample space of a random experiment. In practice, we drop the explicit form X ( ξ ) and simply express the random variable as X . In this course, we take the range of X ( ξ ) to be a subset of the set of all real numbers. Formal Definition A random variable X is a process of assigning a number X ( ξ ) to every outcome ξ . The resulting function must satisfy the
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Unformatted text preview: following two conditions but is otherwise arbitrary: 1 The set { X ≤ x } is an event for every x . (In the applications, we are interested in the probability that a random variable X takes values in a certain region R of the x axis – this mild restriction is mainly of mathematical interest.) 2 The probabilities of the events { X = ∞} and { X =-∞} equal to 0: P { X = ∞} = and P { X =-∞} = (i.e., although we allow X to be ∞ or-∞ for some outcomes, we demand that these outcomes form a set with zero probability.) Example An urn contains 90 $ 1 bills, 9 $ 5 bills, and 1 $ 50 bill. Let the random variable X be the denomination of a bill that is selected at random from the urn. (a) Describe the underlying space S of this random experiment and specify the probabilities of its elementary events. (b) Describe the sample space of X , S X , and find the probabilities for the various values of X ....
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This note was uploaded on 05/05/2010 for the course ECECS 5605 taught by Professor Dasilver during the Fall '08 term at Virginia Tech.

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handout_week2_a - following two conditions but is otherwise...

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