Random variable

We can think of a random variable as a numeric result of operating a non-deterministic mechanism. The mechanism can be as simple as a coin or die to be tossed or a rapid counter which cycles many times in the interval of a typical human physical reaction time until stopped. Mathematically, we can describe it as a function whose domain is a sample space and whose range is some set of numbers.

We can always specify a random variable by specifying its cumulative distribution function because two random variables with identical cdf's are isomorphic. From the cdf, we can calculate probabilities for any events which can be described as countable intersections and unions of intervals.

discrete random variable -- continuous random variable

back to probability distribution

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What about random variables whose range is not numerical? These random variables do not have CDF's. Similarly, while the CDF for a random variable whose range is more than one-dimensional can be defined, it is much more difficult to deal with. -- TedDunning