---- Suppose $(\Omega, \mathcal{F},\mathbb{P})$ is a [[probability|probability space]]. Let $\mathcal{B}$ denote the usual [[Borel set|Borel σ-algebra]] on $\mathbb{R}$. Let $(E, \mathcal{E})$ be a [[σ-algebra|measurable space]]. > [!definition] Definition. ([[probability distribution]]) >Each [[random variable]] $X:\Omega \to E$ determines a [[probability|probability measure]] $\mathbb{P}_X$ on $(E, \mathcal{E})$ by [[pushforward measure|pushing forward]] thus: $\mathbb{P}_{X}(B)=\mathbb{P}(X \in B)=\mathbb{P}\big(X ^{-1} (B)\big).$ >This [[measure]] is called the **(probability) distribution** of $X$. > For two [[random variable|random variables]], defined possibly on different [[probability|probability spaces]], we often write $X \sim Y$ to indicate $\mathbb{P}_{X}=\mathbb{P}_{Y}$. In this case we say $X$ and $Y$ are **identically distributed**. > [!justification]- > We must show $\mathbb{P}_{X}$ is indeed a [[probability|probability measure]]. To begin, $\mathbb{P}_{X}$ is a [[measure]] because $\mathbb{P}_{X}(\bigsqcup_{k=1}^{\infty}E_{k})=\mathbb{P}(X ^{-1} (\bigsqcup_{k} E_{k}))=\mathbb{P}(\bigsqcup_{k} X ^{-1} (E_{k}))=\sum_{k} \mathbb{P}(X ^{-1} E_{k})=\sum_{k} \mathbb{P}_{X}(E_{k}).$ where we have used [[preimages and unions commute]]. And $\mathbb{P}_{X}(\mathbb{R})=\mathbb{P}(X ^{-1}(\mathbb{R}))=\mathbb{P}(\Omega)=1.$ ^justification > [!basicexample] > Suppose $A \in \mathcal{F}$ is an [[probability|event]]. [[indicator random variable|Then]] $\mathbb{P}_{1_{A}}= \big( 1-\mathbb{P}(A)\big) \delta_{0} + \mathbb{P}(A) \delta_{1},$ where $\delta_{0},\delta_{1}$ are [[Dirac measure|Dirac measures]]. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```