# Why are there two ways to write PDF and CDF functions?

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?

• I'd use the subscript when there was any risk it wasn't clear which random variable's density or cdf was intended. May 20 '20 at 22:37

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?