So far, we have considered discrete random variables and their distributions. In applications, such random variables denote the number of objects of a certain type, such as the number of job arrivals to a file server in one minute or the number of calls into a telephone exchange in one minute.

Many situations, both applied and theoretical, require the use of random variables that are “continuous” rather than discrete. As described in the last chapter, a random variable is a real-valued function on the sample space *S*. When the sample space *S* is nondenumerable (as mentioned in Section 1.7), not every subset of the sample space is an event that can be assigned a probability. As before, let denote the class of measurable subsets of *S*. Now if *X* is to be a random variable, it is natural to require that be well defined for every real number *x*. In other words, if *X* is to be a random variable defined on a probability space , we require that be an event (i.e., a member of ). We are, therefore, led to the following extension of our earlier ...

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