In the query complexity model, we define $R(f)$ as the bounded-error query complexity of $f$. It is said that
Claim. we can find two hard distributions over the 0- and 1-inputs of $f$ respectively, such that distinguishing between them with bounded error takes at least $\Omega(R(f))$ time for any randomized algorithm.
I have tried to use Yao's minimax lemma to show this but I failed. More concretely, the obstacle I met is that an algorithm that distinguishes distributions as described need NOT to be a bounded error algorithm for $f$.
Question. How to prove this claim?