Random Variable and Probability Space

Sample Space and Probability Space

Sample Space

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

Probability Measure

A probability measure $P$ on a measurable space $(X,\mathcal{S})$ is a measure that $P(X)=1$.

Probability Space

A probability space $(\Omega ,\mathcal{F},P)$ is a a mathematical construct that provides a formal model of a random process, where

• $\Omega$ is a sample space, which is the set of all possible outcomes.
• $\mathcal{F}$ is an event space, which is a $\sigma$-algebra over $\Omega$.
• $P:\mathcal{F}\to [0,1]$ is a a probability measure, which assigns, to each event a probability.

Random Variable

Random Variable and Probability

Let $(\Omega ,\mathcal{F},P)$ be a probability space, and $(E,\mathcal{E})$ a measurable space. Then an $(E,\mathcal{E})$-valued random variable is a measurable function $X:\Omega \to E$. Then the probability that $X$ takes on a value in a measurable set $S\subseteq E$ is written as$\text{pr}(X\in S)=P\left(\left\{\omega \in \Omega \mid X(\omega )\in S\right\}\right).$