Probability and Statistics

Joint, Marginal, and Conditional

Given $n$ random variables $y_1,...,y_n$.

Joint Probability

Probability of two or more events happening at the same time. $p(y_1,...,y_n)=p(\mathbf{y})$

Marginal Probability

Probability of a single event occurring, without considering any other events.

$$p({\mathbf{y}}_A)=\int p({\mathbf{y}}_A,{\mathbf{y}}_B)d{\mathbf{y}}_B$$ $$p$$

i.e. the integration over the random variable(s) ${\mathbf{y}}_B$ such that the joint distribution $p({\mathbf{y}}_A,{\mathbf{y}}_B)$ collapses into just the marginal distribution of ${\mathbf{y}}_A$.

Conditional Probability

Probability of an event occurring given that another event has already occurred. $p({\mathbf{y}}_A|{\mathbf{y}}_B)=\frac{p({\mathbf{y}}_A,{\mathbf{y}}_B)}{p({\mathbf{y}}_B)}$

Note

If ${\mathbf{y}}_A$ and ${\mathbf{y}}_B$ are independent, then the marginal $p({\mathbf{y}}_A)$ and the conditional $p({\mathbf{y}}_A|{\mathbf{y}}_B)$ are equal.