According to Wikipedia, a random variate is a particular outcome of a random variable, whereas a realization is the observed value of a random variable.
However, it's not clear to me what is the difference between the two. What is the difference and relation between a random variate and realization of a random variable?
Suppose we have a coin toss experiment with sample space $\Omega=\{\text{heads},\text{tails}\}$, with an associated random variable $X$ such that:
$X(\omega )={\begin{cases}1,&{\text{if}}\ \ \omega ={\text{heads}},\\0,&{\text{if}}\ \ \omega ={\text{tails}}.\end{cases}} $
Can I say that the realizations of $X$ are either $1$ or $0$ (before the experiment), and that for an actual observation of the experiment where I tossed heads, $X$ resulted in the random variate $1$? Or, is it the other way around? Or, can I use them interchangeably?
