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2.1 Random Variables and Probability Distributions 61

sample size is sufficiently large, however, the sampling distribution of the sample
average is approximately normal, a result known as the central limit theorem, which
is discussed in Section 2.6.

2.1 Random Variables and Probability

Distributions

Probabilities, the Sample Space, and Random Variables

Probabilities and outcomes. The gender of the next new person you meet, your
grade on an exam, and the number of times your computer will crash while
you are writing a term paper all have an element of chance or randomness. In
each of these examples, there is something not yet known that is eventually
revealed.

The mutually exclusive potential results of a random process are called the
outcomes. For example, your computer might never crash, it might crash once,
it might crash twice, and so on. Only one of these outcomes will actually occur
(the outcomes are mutually exclusive), and the outcomes need not be equally
likely.

The probability of an outcome is the proportion of the time that the outcome
occurs in the long run. If the probability of your computer not crashing while you
are writing a term paper is 80%, then over the course of writing many term papers
you will complete 80% without a crash.

The sample space and events. The set of all possible outcomes is called the sample
space. An event is a subset of the sample space, that is, an event is a set of one or
more outcomes. The event “my computer will crash no more than once” is the set
consisting of two outcomes: “no crashes” and “one crash.”

Random variables. A random variable is a numerical summary of a random
outcome. The number of times your computer crashes while you are writing
a term paper is random and takes on a numerical value, so it is a random
variable.

Some random variables are discrete and some are continuous. As their names
suggest, a discrete random variable takes on only a discrete set of values, like 0, 1,
2, c, whereas a continuous random variable takes on a continuum of possible
values.