# Relationship between "Random Variable" and "Variable of Integration"

I am trying to better understand the relationship between a "Random Variable" and "Variable of Integration" . For example, I was looking at the Tweedie Probability Distribution:

In the above equation, "Y" is the "Random Variable" and "z" is the "Variable of Integration". As far as I understand - from a practical perspective, "Y" and "z" are interchangeable in this equation.

I am struggling to understand the following : If "Y" and "z" are interchangeable in this equation, then why introduce "z" into this equation all together?

• Both terms have definite, clear, standard meanings: apply any definition you are familiar with. They are very different, and so cannot be freely interchanged. One way to see that is to note the "$z$" is not a "variable" at all: it is merely a placeholder in the integral notation and can be changed to any unique symbol you wish without changing the meaning. The best references to consult would be (1) your favorite intro book concerning random variables and (2) your favorite integral Calculus book.
• Consider a simpler distribution: Let $X \sim \mathsf{Beta}(1,0),$ with density function $f_X(x) = 2x,$ for $0<x<1,$ and $0$ elsewhere. Then $E(X) = \int_0^1 xf_X(x)\, dx = \int_0^1 2x^2\, dx = \int_0^1 2\xi^2\, d\xi = \int_0^1 2\aleph^2\, d\aleph = \cdots = 2/3.$ Nov 17 at 19:34