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# exampleclass3solution - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I, FALL 2010 EXAMPLE CLASS 3 Random Variable Elements of Theory A function (with some requirement) defined on the sample space { } is called a random variable. - Domain is the sample space - Range is usually a numbers set, e.g., or its subsets, for easy manipulation. - The range is called the state space of the random variable. There is no intrinsic difference on the nature between a sample space and a state space they are just two sets with some requirement, called “measurability.” They are just domain and range of a “function” with some requireme nt, called “measurability.” The variable perspective is adopted by an observer of a random experiment. The observer is only able to observe/know/measure/obtain information based on the state space. For the observer, all she could see is a variable dancing (randomly) on the state space. This is the perspective that we will primarily study in this course. The function perspective is adopted by someone who would have a “divine” capacity in understanding (a deterministic part of) the design of the random mechanism, in particular, her capacity in seeing the existence of an

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underlying sample space as the domain of a function sending elements of the domain to the state space. You will be studying this perspective in an advanced course of probability. There is a law governing how any random variable to be observed in its state space. The law is a probabilistic one, called the probability distribution of a random variable. There are two qualifications for any real-valued function to be a probability density/mass function, aka, probability function: 1) for any , 2) or .
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