I have the following question: Why do we "swap out" the variable of interest in an integral with some other variable?

For example, in the following integral:

enter image description here

The integral is set up in terms of a variable "Y". However, when the integral is actually written - "Y" is replaced with a new variable "Z" (e.g. dz).

My Question: Does anyone know why this happens? Why is it necessary to do this? Or is this just "mathematical tradition"?



  • 4
    $\begingroup$ $Y$ is a "random variable" - that's a technical term. $z$ is just the variable of integration. These are different concepts. $\endgroup$ Nov 17 '21 at 7:37
  • $\begingroup$ @ ancient mathematician : thank you for your reply! Just a question: is my understanding correct? in the above integral - are "z" and "Y" interchangeable? Thank you! $\endgroup$
    – stats555
    Nov 17 '21 at 7:38
  • 5
    $\begingroup$ I've already said they are different concepts so of course they are not interchangeable. You've really got to go back to the definition of "random variable" and of "probability distribution". $\endgroup$ Nov 17 '21 at 7:40
  • $\begingroup$ Cross-posted: stats.stackexchange.com/questions/552647/… $\endgroup$
    – Peter O.
    Nov 17 '21 at 21:00

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