Welcome to the Statistics page.
Random variable:
This is kinda unedited and not that well written but you get the point. A Random variable is not a variable it is a function.
If we think of X as a “random function” instead we can think of the notation for a regular function f(x). And then think of a random variable as X(x). “F as a function of x or X as a function of x”.
Let’s say the random variable $X_i \sim N(0,1)$
This means that the variable X have the density function: $X(x) = P_X(x) = P(X \leq x) = \int_{-\inf}^{x} \frac{1}{2\sqrt{2\pi\sigma^2}}e^{\frac{(X-\mu)^2}{2\sigma^2}}dx$. Which becomes a number.
And when we write $P(X \leq x)$ . Which reads “whats the probability that $X\leq x$”. In this case it is important to understanding that these x:es are pretty much like in the notation $f(x)$ This is just a somewhat complicated notation.
Another notation for this is $P_{X}(x) = P(X \leq x)$. This is a better notation. If we think of the notation $f(x)$. This is the again same thing.
Let’s say the random variable $X \sim N(0,1)$. This means that the variable X have the density function: $X(x) = P_X(x) = P(X \leq x) = \int_{-\inf}^{x} \frac{1}{2\sqrt{2\pi\sigma^2}}e^{\frac{(X-\mu)^2}{2\sigma^2}}dx$. Which becomes a number.
And when we write $P(X \leq x)$. Which reads “whats the probability that X<x”. In this case it is important to get the understanding that these x:es are not the same. This is just a somewhat complicated notation.
Another notation for this is $P_{X}(x) = P(X \leq x)$. This is a better notation. If we think of the notation $f(x)$. This is the same thing.
So with this in the back of your head. Continue on