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ISEN 609 Lecture 3

ISEN 609 Lecture 3 - We have the concept of a Probability...

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We have the concept of a Probability Space (S, , P ) to describe a random experiment. S and are sets: S is a set of individual outcomes (we can think of them as points). is a set of events (remember that an event is itself a set of outcomes) P is a set function (so we should really think of it as F F F P ( · ) P : F [0 , 1] , or equivalently 0 P ( E ) 1 for each event E F
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A Probability Space is a mathematical construction that allows us to use mathematical properties to model uncertainty. It takes into account the idea that the outcome of a random experiment is not known in advance, but we do know in advance what it could be (sample space) It takes into account that we are usually only interested in certain features or properties of the outcome (events) It takes into account that we want to quantify likelihood on some absolute scale, so we can compare likelihoods between different experiments (probability)
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2. Random Variables A fourth notion, equally important but somewhat less basic, is the idea of a random variable . A random variable is an attribute that can be “measured” from the outcome of the experiment. In mathematical terms, a random variable is actually a function that maps the sample space to the state space of the random variable. The range of the random variable (the collection of values that a random variable can take on), over the domain of outcomes, is called the state space of the random variable. Note: we’ll typically use upper case letters to designate random variables ( X , Y , etc.)
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Random variables Many ( many !) random variables can be defined on the probability space. Random variables are random because they depend on the outcome of an experiment, but they are not variables , but rather functions : Fortunately, the probabilistic structure we’ve developed on the event space easily “transports” to the state space of the random variable.
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