5.1 Random variables

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5.1 Random variables

Thus far, we have been discussing experiments, along with their associated event space, in the context of the probabilities of occurrence of the events. We will now move on to a topic of great importance, which relates the basic probability measures to real-world quantities. The concept of a random variable relates the probabilities of the outcomes of an experiment to a range or set of numbers. A random variable, then, is defined as a function whose input values are the events of the sample space and whose outcome is a real number. For example, we could have an experiment in which the outcome is the length of each message that arrives over a communication line. A random variable defined on this experiment could be the number of messages that equaled a certain character count. Often, we want to consider a range of values of the random variables—for instance, the range of messages greater than x₁. This is denoted here as {X ≤ x₁}, where X denotes the random variable and x₁ is a value for the random variable at a specific point. We may call this set the event where the random variable X yields a value greater than x₁.

Continuing with the previous example, suppose we had the following outcomes from the message-length experiment: The random variable defined by the number of times a particular message length seen. Referring to Figure 5.2; the event {X > 2000} contains the outcomes of messages 1, 4, 5, and 6.

Figure 5.2: Outcomes of the message length experiment.

Random variables may be either discrete or continuous. A discrete random variable is one that is defined on an experiment in which the number of events in the set of outcomes is finite or infinite (i.e., it is possible to assign a positive integer to each event, even if there are an infinite number of outcomes). A continuous random variable is one that is defined on an experiment in which the number of possible outcomes is infinite (i.e., defined on the real line). The concept of random variables forms the foundation for the discussion of probability distributions and density functions.

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