I have a question on the Wiki definition of Variance. Does this form represent abuse of notation or am I interpreting it incorrectly?
The definition: $$ Var(X) = E[(X - E[X])^2] $$ and $$ E[X] = \sum_{i=1}^k c_i p_x(c_i) = \int_{\mathbb{R}} c p_x(c) dc $$
It's my understanding that the parameter of $Var$ and $E$ is identified by a random variable $X$ and the result is a value of the random variable, typically $\in \mathbb{R}$. It seems that the outermost $E$ is given some number other than the symbol for the random variable. I'm also confused by $X - E[X]$, where $X$ appears to be a random variable subtracted by a value $\in \mathbb{R}$.
Appreciate your guidance.
$E[X] = \int x f_X(x) \ dx$ is the expected value of $X$.
You have to imagine that $X$ is a function (it is indeed a function) so what $X-E[X]$ means is just a function minus a real value (which is a function).
For example, if $f(x) = x^2$ you understand what is $f(x) - 2$.
But what you have inside the variance, is taking the expectation of that "function":
\begin{equation*} \begin{split} Var[X] & =E[(X-E[X])^2]=E[X^2 - 2 X E[X] - E[X]^2] \\ & = E[X^2] - 2E[X]^2+E[X]^2 = E[X^2]-E[X]^2. \end{split} \end{equation*} $$$$
If you want to understand everything, you first need to know what a random variable is.
It is a function $$X: \Omega \to E$$ (where usually $E = \mathbb{R}$) with some properties.
So for example, if you consider $E = \mathbb{R}$, when you write $P(X =2)$, this is an abuse of notation for $$P(\{\omega \in \Omega : X(\omega) = 2\})$$ where $P$ is a probability measure and in that case the expectation would be written as $$E[X] = \int_{\Omega} X(w) P(d \omega) = \int_{\Omega} X(w) dP$$
If you want to understand this terms better, I suggest for example that you see the definition of random variable in wikipedia.