\begin{defn}
If $ X_1, \ldots, X_n$ are discrete random variables with $P[X_1 = x_1, X_2 = x_2,\ldots, X_n = x_n] = p(x_1,\ldots, x_n) $ where $x_1 \ldots x_n$ are numbers, then the function $p$ is the joint \textbf{ probability mass function (p.m.f.)} for the random variables $X_1, \ldots, X_n$.
\end{defn}

\begin{defn}
For continuous random variables $Y_1, \ldots, Y_n$, a function $f$ is called the joint probability density function if,

$P [Y\in {A}] = \underbrace{\int\int\ldots\int}_{A} f(y_1,\ldots y_n)dy_1dy_2 \cdots dy_n$.
\end{defn}

\begin{notes}
Note that if $X_1, \ldots, X_n$ are independent and identically distributed, each with p.m.f. $p$, then $p(x_1, x_2, \ldots, x_n) = q(x_1)q(x_2)\ldots q(x_n)$,
i.e, $P [X_1 = x_1, X_2 = x_2,\ldots, X_n= x_n] = P [X_1 = x_1] P[X_2 = x_2]\ldots P[X_n= x_n]$.
\end{notes}

\begin{notes}
Note also that if $A$ is a set of possible outcomes $ (A \subseteq \mathbb{R}^n)$, then
we have
$$P[X \in {A}] = \sum_{(x_1,\ldots,x_n)\in A} p(x_1,\ldots, x_n).$$
\end{notes}

\begin{xmpl}

An urn contains blue and red marbles, which are either light or heavy. Let $X$ denote the color and $Y$ the weight of a marble, chosen at random\\

%\begin{table}[ht]%
%\centering%
\begin{tabular}{c c c c}
\hline\hline
X/Y & L & H & TT \\[0.5ex]
B & 5 & 6 & 11\\
R & 7 & 2 & 9\\
TT & 12 & 8 & 20\\[1ex]
\hline
\end{tabular}
%\end{table}%

We have $P[X="'b"', Y ="l"'] = \frac{5}{20}$.\\

The joint p.m.f. is: \\

%\begin{table}[ht]%
%\centering%
\begin{tabular}{c c c c}
\hline\hline
X/Y & L & H & TT \\[0.5ex]
B & $\frac{5}{20}$ & $\frac{6}{20}$ & $\frac{11}{20}$\\
R & $\frac{7}{20}$ & $\frac{2}{20}$ & $\frac{9}{20}$\\
TT & $\frac{12}{20}$ & $\frac{8}{20}$ & 1\\
\\[1ex]
\hline
\end{tabular}
%\end{table}%
\end{xmpl}