Random Variable

Definition

Random Variable

Let be a probability space and a measurable space of values. A random variable is a function that assigns a value to each outcome and is measurable:

Thus every measurable set of values corresponds to an event whose probability is defined. The function is fixed; its value depends on which outcome occurs.

The value is a realisation of , and the image contains all values can attain. A real-valued random variable uses with its Borel -algebra.

Why Measurability Is Required

The probability measure is defined on events in , not on subsets of the value space . To assign a probability to the condition , collect the outcomes that satisfy it:

Measurability guarantees for every . Hence the probability of is well-defined:

The resulting map

is the probability distribution induced by . Thus a random variable transfers probability from sets of raw outcomes to sets of values.