Random variables are usually denoted with upper-case letters. For example, there could be a random variable $X$. Now, because vectors are usually denoted with a bold lower-case letter (e.g. $\mathbf{z} = (z_0, \dots, z_{n})^{\mathsf{T}}$ and matrices with a bold upper-case letter (e.g. $\mathbf{Y}$), how should I denote a vector of random variables? I think $\mathbf{x} = (X_0, \dots, X_n)^\mathsf{T}$ looks a bit odd. On the other hand if I see $\mathbf{X}$ I would first think it is a matrix. What is the usual way to do this? Of course, I think it would be best to state my notation somewhere in the beginning of paper.
Solved – Notation for random vectors
matrixnotationrandom variable
Related Solutions
There is no single answer to this question because different authors may use different notation. For me, the most handy notation is the one used, for example, by Larry Wasserman in All of Statistics:
By convention, we denote a point estimate of $\theta$ by $\hat\theta$ or $\widehat\theta_n$. Remember that $\theta$ is a fixed, unknown quantity. The estimate $\hat\theta$ depends on the data so $\hat\theta$ is a random variable.
More formally, let $X_1,\dots,X_n$ be $n$ iid data points from some distribution $F$. A point estimator $\widehat\theta_n$ of a parameter $\theta$ is some function of $X_1,\dots,X_n$:
$$ \widehat\theta_n = g(X_1,\dots,X_n). $$
So $\theta$ is the unknown parameter, $\hat\theta$ is the estimate, and a function $g$ of the sample is the estimator. Such notation makes it also clear that $g$ is a function.
The realization of a random variable is the value that was observed (though, as noticed in the comments, you can have random variables for non-observable things). For example, you treat the result of throwing a fair dice as a random variable $X$. Say that the result is five dots, $x=5$ is the realization. The “five objects” that you call “realizations” are all random variables that together form multivariate random variable. In this framework, it doesn’t make sense to discuss a single random variable with multiple realizations.
You can throw a dice $n$ times and treat the results as $X_1,X_2,\dots,X_n$ random variables with $n$ observed realizations accordingly. $E[X_1]$ could be an expected value of the random variable for the result of first throw $X_1$, where the realization $x_1$ would be a number, for example, $3$. So
$$ \bar X = \frac{1}{n} \sum_{i=1}^n X_i $$
is a random variable, as a function of $n$ random variables, and in
$$ \bar x = \frac{1}{n} \sum_{i=1}^n x_i $$
$\bar x$ is a realization of $\bar X$ calculated from realizations $x_i$ random variables $X_i$.
A random vector $\mathbf{X} = (X_1,X_2,\dots,X_n)$ is just a shorthand for writing them all each time.
Finally, you would see different notations used by different authors and in different contexts, so each time you need to make sure what is described rather than assuming things from notation alone.
You should probably refresh your knowledge on random variables to make things clearer. Given the multiplicity of issues mentioned by you, I'd recommend also a probability and statistics handbook or leactures.
Best Answer
The convention to specify vectors or matrices with bold letters is much more frequently upheld than the convention of upper-case letters for random variables. In the articles I usually read (econometrics, time-series regression mostly) the latter convention is not used, i.e. the random variables are usually lower-case.
Look for the influential papers in your field and try to copy their conventions. Stating the notation somewhere in the beginning is a must usually.