$\newcommand{\br}{\\}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\N}{\mathbb{N}}$
$\newcommand{\C}{\mathbb{C}}$
$\newcommand{\P}{\mathbb{P}}$
$\newcommand{\F}{\mathbb{F}}$
$\newcommand{\L}{\mathcal{L}}$
$\newcommand{\spa}[1]{\text{span}(#1)}$
$\newcommand{\dist}[1]{\text{dist}(#1)}$
$\newcommand{\max}[1]{\text{max}(#1)}$
$\newcommand{\min}[1]{\text{min}(#1)}$
$\newcommand{\supr}[1]{\text{sup}(#1)}$
$\newcommand{\infi}[1]{\text{inf}(#1)}$
$\newcommand{\ite}[1]{\text{int}(#1)}$
$\newcommand{\ext}[1]{\text{ext}(#1)}$
$\newcommand{\bdry}[1]{\partial #1}$
$\newcommand{\argmax}[1]{\underset{#1}{\text{argmax }}}$
$\newcommand{\argmin}[1]{\underset{#1}{\text{argmin }}}$
$\newcommand{\set}[1]{\left\{#1\right\}}$
$\newcommand{\emptyset}{\varnothing}$
$\newcommand{\tilde}{\text{~}}$
$\newcommand{\otherwise}{\text{ otherwise }}$
$\newcommand{\if}{\text{ if }}$
$\newcommand{\proj}{\text{proj}}$
$\newcommand{\union}{\cup}$
$\newcommand{\intercept}{\cap}$
$\newcommand{\abs}[1]{\left| #1 \right|}$
$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$
$\newcommand{\pare}[1]{\left(#1\right)}$
$\newcommand{\brac}[1]{\left[#1\right]}$
$\newcommand{\t}[1]{\text{ #1 }}$
$\newcommand{\head}{\text H}$
$\newcommand{\tail}{\text T}$
$\newcommand{\d}{\text d}$
$\newcommand{\limu}[2]{\underset{#1 \to #2}\lim}$
$\newcommand{\der}[2]{\frac{\d #1}{\d #2}}$
$\newcommand{\derw}[2]{\frac{\d #1^2}{\d^2 #2}}$
$\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}}$
$\newcommand{\pderw}[2]{\frac{\partial^2 #1}{\partial #2^2}}$
$\newcommand{\pderws}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}}$
$\newcommand{\inv}[1]{{#1}^{-1}}$
$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$
$\newcommand{\nullity}[1]{\text{nullity}(#1)}$
$\newcommand{\rank}[1]{\text{rank }#1}$
$\newcommand{\nullspace}[1]{\mathcal{N}\pare{#1}}$
$\newcommand{\range}[1]{\mathcal{R}\pare{#1}}$
$\newcommand{\var}[1]{\text{var}\pare{#1}}$
$\newcommand{\cov}[1]{\text{cov}(#1)}$
$\newcommand{\cov}[2]{\text{cov}\pare{#1, #2}}$
$\newcommand{\tr}[1]{\text{tr}(#1)}$
$\newcommand{\oto}{\text{ one-to-one }}$
$\newcommand{\ot}{\text{ onto }}$
$\newcommand{\ceil}[1]{\lceil#1\rceil}$
$\newcommand{\floor}[1]{\lfloor#1\rfloor}$
$\newcommand{\Re}[1]{\text{Re}(#1)}$
$\newcommand{\Im}[1]{\text{Im}(#1)}$
$\newcommand{\dom}[1]{\text{dom}(#1)}$
$\newcommand{\fnext}[1]{\overset{\sim}{#1}}$
$\newcommand{\transpose}[1]{{#1}^{\text{T}}}$
$\newcommand{\b}[1]{\boldsymbol{#1}}$
$\newcommand{\None}[1]{}$
$\newcommand{\Vcw}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Vce}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Vcr}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Vct}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Vcy}[6]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{bmatrix}}$
$\newcommand{\Vcu}[7]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{bmatrix}}$
$\newcommand{\vcw}[2]{\begin{matrix} #1 \br #2 \end{matrix}}$
$\newcommand{\vce}[3]{\begin{matrix} #1 \br #2 \br #3 \end{matrix}}$
$\newcommand{\vcr}[4]{\begin{matrix} #1 \br #2 \br #3 \br #4 \end{matrix}}$
$\newcommand{\vct}[5]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \end{matrix}}$
$\newcommand{\vcy}[6]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{matrix}}$
$\newcommand{\vcu}[7]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{matrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Mqr}[4]{\begin{bmatrix} #1 & #2 & #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqt}[5]{\begin{bmatrix} #1 & #2 & #3 & #4 & #5 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mrq}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Mtq}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Mww}[4]{\begin{bmatrix} #1 & #2 \br #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mwe}[6]{\begin{bmatrix} #1 & #2 & #3\br #4 & #5 & #6 \end{bmatrix}}$
$\newcommand{\Mew}[6]{\begin{bmatrix} #1 & #2 \br #3 & #4 \br #5 & #6 \end{bmatrix}}$
$\newcommand{\Mee}[9]{\begin{bmatrix} #1 & #2 & #3 \br #4 & #5 & #6 \br #7 & #8 & #9 \end{bmatrix}}$
Definition: Joint PMFs
Let $\b{X} = (X_1, X_2, …, X_n)$ be a vector of $n$ discrete random variables,. Let $\b{x} = (x_1, x_2, …, x_n) $ be possible values of the random variables, the PMF of the event
$$\set{ \b{X} = \b{x}} = \set{X_1 = x_1, X_2 = x_2, …, X_n = x_n}$$
or more precisely
$$\bigcap_{i=1}^n \set{X_i = x_i}$$
is the joint PMF, denoted as follow
$$p_{ \b{X}}(\b{x}) = \b{P}(\b{X} = \b{x})$$
The joint PMF determines the probability of any event $A$ that can be specified in terms of the random variables $X_1, X_2, …, X_n$
$$\b{P}(\b{X} \in A) = \sum_{ \b{X} \in A } p_{ \b{X}}(\b{x})$$
Definition: Marginal PMFs
Given a joint PMF $p_ \b{X}$, we call $p_{X_i}(x_i), i \in \set{ 1, 2, …, n }$ the marginal PMF
$$p_{X_i}(x_i) = \sum_{X_1} … \sum_{X_{i-1}}\sum_{X_{i+1}}… \sum_{X_n} p_{\b{X}} (\b{x})$$
Definition: Functions of Multiple Random Variables
Let $X_1, X_2, …, X_n$ be $n$ discrete random variables denoted as $\b{X}$. $Y$ be another random variable defined as $Y = g(\b{X})$. Its PMF can be calculated from the joint PMF according to
$$p_Y(y) = \sum_{\set{ \b{x} | g(\b{x}) = y}} p_{ \b{X}}(\b{x})$$
The expected value rule for functions naturally extends and takes the forms
$$E [g(\b{X})] = \sum_{X_1} \sum_{X_2}… \sum_{X_n} g(\b{x}) p_{\b{X}}(\b{x})$$