I am currently reading Chapter 7 of Stephen Boyd's textbook. From Stephen Boyd's optimization textbook, it says
This is already very confusing;
From my understanding,
If $X$ is a random variable on a probability space $(\Omega, \mathcal{A}, \mu)$, then $X$ induces a probability measure on $\mathbb{R}$ called the probability distribution, $\mu(A) = \Pr(X \in A)$, $A \in \mathcal{A}$.
A distribution function of $X$ is the function $F(x) = \Pr(X\leq x)$, also known as the CDF.
The density of $F$ is the function $f$ such that $F(x) = \int\limits_{-\infty}^x f(y) dy$
So what does probability distribution "indexed" by a vector mean in this context? Also I am troubled by the notation $p_x(\cdot)$. In my experience, almost all densities are written as $f_X(x)$, where $X$ is the random variable. But here $x$ is just a vector. In the entire chapter, the notion of a random variable is never even brought up, yet curiously a probability distribution etc is still able to be defined.
Can someone help me understand the meaning of the very first sentence and offer an example of what a concrete probability distribution and density $p_x(\cdot)$ in this context?