> [!definition] > > Let $(\Omega, \cf, \bp)$ be a [[Probability|probability space]] and $M$ be a [[Topological Space|topological space]]. A mapping $X: \Omega \to M$ is a $M$-valued **random variable** if $X$ is $(\cf, \cb(\cm))$-[[Measurable Function|measurable]] ([[Borel Measurable Function|Borel measurable]]). > > A **real**-valued random variable is simply known as a random variable. > [!definition] > > Let $(\Omega, \cf, \bp)$ be a probability space and $X$ be a random variable. The $\sigma$-field generated by $X$ is the smallest [[Sigma Algebra|sigma algebra]] such that $X$ is measurable: > $ > \sigma(X) = \sigma\bracs{X^{-1}(B): B \in \cb(\real)} > $ > [!definition] > > Let $(\Omega, \cf, P)$ be a [[Probability|probability]] space. A random variable $X$ is a real-valued [[Function|function]] from $\Omega$ to $\real$ that assigns a numerical value to each outcome. > > For any subset $S \subseteq \real$, denote > $ > P(X \in S) = P(X^{-1}(S)) > $ > as the probability for $X$ to take values in $S$. > [!definition] > > Let $X$ be a random variable on a probability space $(\Omega, \cf, P)$. $X$ is a **discrete** random variable if there exists distinct real numbers $\seq{x_i}$[^1] such that > $ > \sum_{i \in \nat}P(X = x_i) = 1 > $ > [!definition] > > Let $X$ be a random variable on a probability space $(\Omega, \cf, P)$. $X$ is a **continuous** random variable if there exists an [[Integrable Function|integrable function]] $f: \real \to \real$ such that > $ > P(X \in B) = \int_B f(t)dt \quad \forall B \in \cb > $ > Any two such functions are equal [[Almost Everywhere|a.e.]]