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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:

enter image description here

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?

Can someone please help me understand this?

Thanks!

References:

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    $\begingroup$ 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. $\endgroup$
    – whuber
    Nov 17 at 18:27
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    $\begingroup$ This content requires some familiarity with measure theory from a reader, just a warning $\endgroup$
    – Aksakal
    Nov 17 at 18:33
  • $\begingroup$ Thank you everyone for your replies! $\endgroup$
    – stats555
    Nov 17 at 18:40
  • $\begingroup$ 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 $E(X) = \int_0^1 xf_X(x)\, dx = \int_0^1 2x^2\, dx = \int_0^1 2\xi^2\, d\xi = \int_0^1 2\aleph^2\, d\aleph = \cdots = 2/3.$ $\endgroup$
    – BruceET
    Nov 17 at 19:34
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    $\begingroup$ Cross-posted: math.stackexchange.com/questions/4308439/… $\endgroup$
    – Peter O.
    Nov 17 at 21:00

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