$Y_{(n)} = X_{(n)}/\mu$?

Question

If $ X_1, ...,X_n$ are iid random variables such that $ X_i \sim U(0, \mu)$, is that true that if $Y_i = X_i/\mu$, then $Y_{(n)} = X_{(n)}/\mu?$

I am sorry if the question looks so simple and I am nt giving any attempt of answer, but intuitively, I think the affirmation is true, thinking on properties of the maximum. But I am not sure about it, since we are talking about random variables and they are functions.

Thanks in advance!

Answer 1

$$X_{(n)}=\max X_i\implies X_{(n)}/\mu=\max X_i/\mu=\max Y_i\qquad (\mu\gt0)$$

Answer 2

If $\mu>0$ and they are functions on - let's say $\Omega$ - then by definition for $\omega\in\Omega$: $$Y_{(n)}(\omega)=\max\{Y_1(\omega),\dots,Y_n(\omega)\}=\max\{X_1(\omega)/\mu,\dots,X_n(\omega)/\mu\}=\max\{X_1(\omega),\dots,X_n(\omega)\}/\mu=X_{(n)}(\omega)/\mu$$