I'm getting rather confused with the random variable concept and its distribution in probability, especially when it gets abstract with no actual example to base my understanding on.
Take for example a sample space where $\Omega = \left \{ \omega _{1} ,\omega _{2}, \omega _{3} \right \}$ and $\mathbb{P}(\omega _{1})=\frac{1}{2}$, $\mathbb{P}(\omega _{2})=\mathbb{P}(\omega _{3})=\frac{1}{4}$. $X, Y, Z$ are defined as such:
$X(\omega_{1}) = 1$ , $X(\omega_{2}) = 2$ , $X(\omega_{3}) = 2$
$Y(\omega_{1}) = 2$ , $Y(\omega_{2}) = 1$ , $Y(\omega_{3}) = 1$
$Z(\omega_{1}) = 1$ , $Z(\omega_{2}) = 2$ , $Z(\omega_{3}) = 1$
To show: That $X$ and $Y$ have the same distribution.
The question is then, how do I intuitively interpret the whole idea? And how would the probability of $X, Y$ or $Z$, or even say, where arithmetic operations are used, $X+Y, XY$, be understood?
(My idea is that the distribution of $X$ is simply defined by the $\mathbb{P}(\omega _{1})X(\omega_{1}) + \mathbb{P}(\omega _{2})X(\omega_{2}) + \mathbb{P}(\omega _{3})X(\omega_{3})$, but as to whether that's true or why it is true, if true, I have no clue. )
Thanks!