What does the subscript of a CDF mean?

Question

I am working with the composition method for generating a random variable $X$. I have always seen CDFs denoted as $F_X(x)$ but my question is what does it mean if the CDF is $F_I(X)$. So specifically what does the change from $X$ to $I$ imply?

Edit: I flipped the X and x in my original post.

Answer 1

I think you swaped uppercase and lowercase $X$.

By definition for a given random variable $X$ you can evaluate the CDF $F_X$ at the point $x$ : $$F_X(x) = \mathbb{P} (X \leq x)$$

If you want to calculate the CDF of an other random variable $Y$ the expression becomes : $$F_Y(x) = \mathbb{P} (Y \leq x)$$

Answer 2

I have NEVER seen $F_x(X),$ but always only $F_X(x).$ The capital $X$ identifies which random variable it is; the lower-case $x$ is the argument to the function. Thus $F_X(3) = \Pr(X\le 3)$ and $F_Y(3) = \Pr(Y\le 3).$

Without this distinction between capital $X$ and lower-case $X,$ how would one even understand something like $\Pr(X\le x)\text{?}$