Probability
The Analysis of Data, volume 1
Random Vectors: Basic Definitions
$
\def\P{\mathsf{\sf P}}
\def\E{\mathsf{\sf E}}
\def\Var{\mathsf{\sf Var}}
\def\Cov{\mathsf{\sf Cov}}
\def\std{\mathsf{\sf std}}
\def\Cor{\mathsf{\sf Cor}}
\def\R{\mathbb{R}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\mathsf{\sf diag}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
$
4.2. Joint Pmf, Pdf, and Cdf Functions
Definition 4.2.1.
The random vector $\bb{X}$ is discrete if there exists a finite or countable set $K\subset\R^n$ such that $\P(\bb{X} \in K)=1$. The random vector $\bb{X}$ is continuous if $\P(\bb{X}=\bb{x})=0$ for all $\bb{x}\in\mathbb{R}^n$.
The random vector in Example 4.1.1 may be continuous (assuming weight, height and IQ are measured with infinite precision) while the random vector in Example 4.1.2 is clearly discrete. As with one dimensional random variables, we define the cdf, the pmf, and the pdf below.
Definition 4.2.2.
For any random vector $\bb{X}=(X_1,\ldots,X_n)$ we define the cdf as $F_{\bb X}:\R^n\to \R$
\[F_{\bb{X}}(\bb{x})=\P(X_1\leq x_1,X_2\leq x_2,\ldots,X_n\leq
x_n).\]
For continuous $\bb X$ we define the pdf as
\[f_{\bb{X}}(\bb{x})
=\frac{\partial^n}{\partial x_1\cdots \partial
x_n}F_{\bb{X}}(x_1,\ldots,x_n)\]
and 0 if the derivative does not exist. For discrete $\bb X$ we define the pmf as
\[p_{\bb{X}}(\bb{x})=\P(X_1=x_1,\ldots,X_n=x_n).\]
The cdf, pdf, and pmf of a random vector have similar properties to the properties of the cdf, pdf, and pmf of a random variable derived in Chapter 2 (with obvious modification). Most proofs carry over from the random variable case with little modifications. Examples appear below.
- The pdf is non-negative and integrates to one \[\idotsint_{\R^n} f_{\bb{X}}(\bb{x}) \,dx_1\cdots dx_n=1.\]
- The pmf is non-negative and sums to one \[\sum_{x_1}\cdots\sum_{x_2} p_{\bb{X}}(\bb{x})=1.\]
- The cdf is monotonic increasing: if $a_i\leq b_i$ for all $i=1,\ldots,n$, then
\[F_{\bb{X}}(a_1,\cdots,a_n)\leq F_{\bb{X}}(b_1,\cdots,b_n).\]
-
\begin{align*}
0 &= \lim_{x_1\to-\infty}\cdots \lim_{x_n\to-\infty} F_{\bb{X}}(x_1,\cdots,x_n)\\
1 &= \lim_{x_1\to\infty}\cdots \lim_{x_n\to\infty} F_{\bb{X}}(x_1,\cdots,x_n).
\end{align*}
- For continuous random vectors
\[F_{\bb{X}}(\bb x) = \int_{-\infty}^{x_1}\int_{-\infty}^{x_2}
\cdots \int_{-\infty}^{x_n}
f_{\bb{X}}(t_1,t_2,\ldots,t_n) \,\,dt_1 dt_2 \cdots dt_n.\]
-
\begin{align}
\P(\bb{X}\in A)=\begin{cases} \sum_{\bb{x}\in A}
p_{\bb{X}}(\bb{x}) & \bb{X} \text{ is a discrete
random vector}\\
\int_A f_{\bb{X}}(\bb{x})\,
d\bb{x} & \bb{X} \text{ is a continuous random vector}
\end{cases}.
\end{align}