Different CDF Notations

Question

I understand that $F_X(x)$ is the CDF of the random variable X and $F_Y(y)$ is the CDF of the random variable Y.

What do these mean? 1. $F_X(y)$ 2. $F_X(X)$ 3. $F_x(Y)$

Answer 1

$F_X(y)$ is the value of $P\{X \leq y\}$, the probability that $X$ is no larger than the real number $y$.

$F_X(X)$ is a function of the random variable $X$ and is thus also a random variable; call it $Z$. It is an interesting fact that if $F_X(x)$ is a continuous strictly increasing function of $x$, then $Z$ is a uniformly distributed random variable on the interval $(0,1)$.

$F_x(Y)$ is a function of the random variable $Y$ and thus also a random variable but you have not provided a definition of the function $F_x$ and so there is not much more that can be said. Perhaps you meant $F_X(Y)$? Well, there is not much more that can be said except that $F_X(Y)$ also takes on values in $(0,1)$ or $[0,1)$ or $(0,1]$ or $[0,1]$ depending on what the function $F_X$ looks like.

Answer 2

$F_X$ and $F_Y$ are just short names that refer to mathematical expressions. You can pretty much always plug a number into an expression, even if doing so doesn't make sense. The meaning of the expression doesn't say anything about what you can plug into it.*

$F_X$ describes the distribution of $X$. It's also just a function like any other; it just happens to be associated with $X$. Think of another example: given the perimeter of a circle (denoted $p_C$), you can compute the area with the function $$ a_C{\left(p_C\right)} = \pi \left(\frac{p_C}{2\pi}\right)^2 = \frac{p_C^2}{4\pi} $$ And you can compute the area of a square from the perimeter (denoted $p_S$) with $$ a_S{\left(p_S\right)} = \left(\frac{p_S}{4}\right)^2 = \frac{p_s^2}{16} $$

In these expressions $p_C$ and $p_S$ are really just placeholders. They could just as easily have been called $\aleph$ and $\beth$. Using $p_C$ and $p_S$ just makes it clear that the intended inputs are the perimeter of a circle and the perimeter of a square, respectively.

The thing is that you can't get too hung up on those placeholders. $a_C{\left(\cdot\right)}$ is only meaningful when $\cdot$ is the perimeter of a circle. But it always has a value, whether $\cdot$ is the perimeter of a square, the speed of light, or the number of words in this answer.

*This isn't always true, but it is close enough to true that you can use it to fix the idea.

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So to answer the first question: $F_X(y)$ is the expression denoted $F_X$ (which just happens to be the CDF of $X$) evaluated at some value $y$.

The second question has an added level of complexity. It is conventional to use capital letters ($X$) to refer to random variables, and lower-case letters ($x$) to refer to realizations of random variables. Realizations are just numbers, so $F_X(x)$ is just a number (in this case, a probability). But the result of applying a function to a random variable, as in $F_X(X)$, is actually another random variable. The distinction is easy to see if you define $U = F_X(X)$. $U$ is a random variable, and every realization $u$ is equal to some $F_X(x)$. So $F_X(X)$ is the random variable produced by applying $F_X$ to $X$.

The same principle applies to the third question. $F_X(Y)$ is the random variable produced by applying $F_X$ to $Y$. All the stuff up there about squares and circles applies in this case, too.

Incidentally, my first ever question on this site was about the distribution of $F_X(X)$.