Random Variables

Definition (Measurable)

If and are two measurable spaces, a function is called measurable if for any set , the set

is in , i.e., for any set , there is a set in which maps.

Statistics and data mining are concerned with data. How do we link sample spaces and events to data? The link is provided by the concept of a random variable.

Definition (Random variable)

A random variable is a measurable function from a probability space to the reals, i.e.,

that assigns a real number to each outcome .

Given a random variable and a subset of the real line, define , and let

Notice that denotes the random variable, and denotes a particular value of .

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 2.