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I often see PDF and CDF functions written as either

$f_X(x)$ or $f(x)$ for PDF

or

$F_X(x)$ or $F(x)$ for CDF.

In what situations would you use either notation? Like what is the point of having two ways?

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    $\begingroup$ I'd use the subscript when there was any risk it wasn't clear which random variable's density or cdf was intended. $\endgroup$
    – Glen_b
    May 20 '20 at 22:37
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The capital letter means the random variable for which the function if the CDF or PDF. If you're just dealing with $X$ and $Y$ as the random variables, it is easy to write $f(x)$ and $g(y)$ and drop the subscript; it is clear from the context that we mean the PDF (ditto for $F(x)$ and $G(y)$ being CDFs) of $X$ and $Y$.

When you have many random variables, you run out of letters. In that case, it is easiest to call all of the random variables with subscripts, like $X_1, X_2,\dots, X_k$. In that case, we would denote the PDF as $f_{X_1}(x_1)$, etc. I suppose that we could call it $f_1(x_1)$ and not cause confusion, but why not include the whole random variable?

It's fine to use the shorthand notation that omits the subscript when the context makes it clear what you mean...but only if the context makes it clear what you mean.

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