Does the inverse of each element of a sample from a random variable constitute a sample from the inverse of that variable?

Question

Imagine you have a sample of $n$ elements from a random variable $X$. If I take the inverse of every element in that sample, is it always the case that the resulting set of numbers constitute a sample from $\frac{1}{X}$ ?

Thanks in advance

Answer 1

Yes.
The inverses remain independent.
The rest of it is just thinking through what exactly it means for a random variable to have the distribution of the random variable $\frac{1}X$.