Random Variable and Probability Space

Sample Space and Probability Space

Sample Space

The sample space of an experiment is the set of all possible outcomes of the experiment. An event is a subset of the sample space , and we say that occurred if the actual outcome is in .

Probability Measure

A probability measure on a measurable space is a measure that .

Probability Space

A probability space is a a mathematical construct that provides a formal model of a random process, where

• is a sample space, which is the set of all possible outcomes.
• is an event space, which is a -algebra over .
• is a a probability measure, which assigns, to each event a probability.

Random Variable

Random Variable and Probability

Let be a probability space, and a measurable space. Then an -valued random variable is a measurable function . Then the probability that takes on a value in a measurable set is written as