Multinomial
The multinomial is an example of a parametric distribution for multiple random variables.
Say you perform $n$ independent trials of an experiment where each trial results in one of $m$ outcomes, with respective probabilities: $p_1, p_2, \dots , p_m$ (constrained so that $\sum_i p_i = 1$). Define $X_i$ to be the number of trials with outcome $i$. A multinomial distribution is a closed form function that answers the question: What is the probability that there are $c_i$ trials with outcome $i$. Mathematically: \begin{align*} P(X_1=c_1,X_2 = c_2, \dots , X_m = c_m) = { {n} \choose {c_1,c_2,\dots , c_m} }\cdot p_1^{c_1} \cdot p_2^{c_2}\dots p_m^{c_m} \end{align*}
Often people will use the product notation to write the exact same equation: \begin{align*} P(X_1=c_1,X_2 = c_2, \dots , X_m = c_m) = { {n} \choose {c_1,c_2,\dots , c_m} }\cdot \prod_i p_i^{c_i} \end{align*}
The multinomial is especially popular because of its use as a model of language. For a full example see the Federalist Paper Authorship example.