Is Z a random variable of X if it is transformed from X? If Z is a random variable, and therefore a function, of X, could one denoted Z as Zₓ(x)?
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2I haven't come across symbols like Zₓ(x) that are used conventionally to mean $Z = f(X)$ if that is what you are implying. – User1865345 Dec 30 '22 at 14:52
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Is your concern that $Z$ might not be so random if $Z$ is determined by $X?$ – Dave Dec 30 '22 at 15:32
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4One would ordinarily write $Z(X)$ for the transformation. The subscript you propose is superfluous and highly unusual. Your initial question is answered at https://stats.stackexchange.com/a/94876/919. – whuber Dec 30 '22 at 15:35
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2Does this answer your question? Functions of Independent Random Variables – jbowman Dec 30 '22 at 19:26
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thank you, guys – Joseph_and_Betty Dec 31 '22 at 02:40
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Let's flip it around for clarify.
If $X$ is a random variable, then all the common transformations of $X$ are also random variables. However, these transformations likely do not have the same distribution as $X$. For example, if $Z = e^X$ and $X$ is normal distributed, then $Z$ is log-normally distributed (literally, $\ln(Z)$ is normal). Other transformations have other relationships.
Note, some functions of $X$ do not give a random variable. For example, $\frac{X}{X}$ would of course always equal 1 (when $X\neq 0$).
As for notation, you can just use $Z(X)$. No need to the subscript. Although, if if you have several variables to transform, e.g. $X$ and $Y$, then the subscript could be useful when you are evaluating specific values: e.g. $Z_X(4)$ vs $Z_Y(4)$.
User1865345
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Michael Cooney
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1In order to avoid constantly caveating our statements with trivial exceptions, we allow that $X/X$ is a random variable even though it is either constant or undefined. When there's a positive chance that $X=0,$ then $X/X$ is not defined at all. The functions of $X$ that give a random variable are the measurable functions and, again, any constant function is (Borel) measurable. – whuber Dec 31 '22 at 00:52
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2@Michael Where you write "when $X\neq 1$" did you mean to write "X\neq 0$" instead? – Glen_b Dec 31 '22 at 01:08
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