Yes, that’s important clarification. Markov’s inequality is tight on the space of all non-negative random variables (the inequality becomes an equality with the two-point distribution shown in the final state of the proof). But it’s not constructed to be tight with respect to a generic distribution.
I’m pretty new to these sorts of tail-bound proofs that you see a lot in e.g high-dimensional probability theory. But in general, understanding under what circumstances a bound is tight has been one of the best ways to intuitively understand how a given bound works.
Yes, that’s important clarification. Markov’s inequality is tight on the space of all non-negative random variables (the inequality becomes an equality with the two-point distribution shown in the final state of the proof). But it’s not constructed to be tight with respect to a generic distribution.
I’m pretty new to these sorts of tail-bound proofs that you see a lot in e.g high-dimensional probability theory. But in general, understanding under what circumstances a bound is tight has been one of the best ways to intuitively understand how a given bound works.