Change of variable for integral formula...why?

Question

I'm reading a book on mathematical statistics and it states two properties of pdfs.

1) $f_X(x)\geq 0$
2) $\int_{-\infty}^{\infty}f_X(t)dt = 1$

My question is this. Why is the integration variable a t, not an x in the second equation?

Thanks

Answer 1

This is only to close the question.

Absolutely no mathematical motivation. Probably to remove confusion with the $X$. But remember , that choice of notation, especially while writing a book, must be very careful, since one requires it to be consistently maintained throughout the book.

Therefore, this notation may by repeated through the book : as you read on, you will realize the role of t instead of x.

If notation is cluttered or variables are not chosen nicely, you could land in trouble. For example, I could write $f_X(t)$ as $\mathcal F_F(f)$ and there would be confusion as to what is what.

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I should add that $t$ is varying from $-\infty$ to $\infty$. My guess is that $t$ usually varies over the reals in the book, as an integration variable. I have seen this in other books on probability as well.