---- > [!definition] Definition. ([[marginal distribution]]) > Consider a [[probability|probability measure]] $\mathbb{P}$ on a [[product of σ-algebras|product]] of [[measure|probability spaces]] $(\Omega, \mathcal{F})=\left(\prod_{i \in I}^{}\Omega_{i}, \bigotimes_{i \in I}\mathcal{F}_{i}\right)$. Denote by $\mathbb{P}_{i}:\mathcal{F}_{i} \to [0,1]$ the $i$th [[product of σ-algebras|marginal measure]] of $(\Omega, \mathcal{F})$; recall this is given by $\mathbb{P}_{i}(A)=\mathbb{P}\left( A \times \prod_{j \neq i}^{} \Omega_{j} \right) \text{ for an event }A \in \mathcal{F}_{i}.$ We call $\mathbb{P}_{i}$ the **$i$th marginal distribution** of $(\Omega, \mathcal{F}, \mathbb{P})$. Its the [[probability distribution]] of a (generally not real-valued) [[random variable]] $\prod_{i \in I}^{}\Omega_{i} \to \Omega_{i}$. > The marginals $\mathbb{P}_{i}$ do not determine $\mathbb{P}$. > [!intuition] > The marginal distribution describes the likelihood of an [[event]] to occur, irrespective of others. Marginal variables are so named because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. > ![[CleanShot 2023-01-15 at 14.13.54.jpg|200]] > [!basicexample] > Suppose we have a continuous [[joint probability distribution|random vector]] $\v X=(X_{1},X_{2})$ and a [[random variable]] $Y$ that takes on values in $\{ 1,2 \}$. When we say "the [[marginal distribution]] of $Y$ is uniform on the values $1$ and $2quot;, we are saying that $P(Y=1)=\frac{1}{2} \and P(Y=2)=\frac{1}{2}$. ---- #### ---- #### 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 > ```