# What is a sampling procedure for $T(X)$ given the sampling procedure for $X$ where $T$ is a deterministic map?

Given $$X$$ is a random variable and $$Y = T(X)$$ where $$T$$ is a deterministic function, then, $$Y$$ is a random variable. Consider the following sampling procedure: sample $$x \sim X$$, then apply $$T$$ on $$x$$ to get $$T(x)$$. Is this sampling procedure guaranteed to be that for sampling from $$Y = T(X)$$?

This question could be trivial (because it is intuitively correct that the sampling procedure above is indeed for $$Y$$) but I could not either find a counter-example against that or find some text that explicitly claims that.

Thank you.

• Yes, $T(x)$ follows the distribution of $Y$ by definition. Jul 14, 2019 at 7:22
• @angryavian Do you have any pointer to that definition?
– TNg
Jul 14, 2019 at 7:35

I agree with you and angryavian that the statement is trivially correct, since for every $$A$$ in the considered $$\sigma$$-algebra we have $$P(T(x)\in A) \stackrel{x\sim X}{=} P(T(X)\in A) = P(Y\in A).$$