Random variables

Real-valued random variables

In this case the observation space is the set of real numbers. Recall, ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ is the probability space. For a real observation space, the function ${\displaystyle X\colon \Omega \rightarrow \mathbb {R} }$ is a real-valued random variable if $${\displaystyle \{\omega :X(\omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R} .}$$