Let $f: D \rightarrow D$ be a permutation, and suppose $X$ is uniformly distributed, i.e. $X$ is a uniform random variable with support $D$.
Then $f(X)$ is also a random variable with support $D$, and also uniform. How would I formally show or explain that $f(X)$ is uniform?
Does the cardinality of $D$ matter at all here? Are permutations only defined on countable sets?
Since all values have the same probability, permuting them can't change this value . E.g. if 2 and 3 are permuted , the probability of 2 becomes the probability of 3 , but they are the same, so identical with the original value .
The cardinality matter in computing the probability of each value ( which is the same for all values , for the uniform distribution).
Permutations on uncountable sets does not make much sens .
