This two-point distribution is important as the distribution where Markov’s inequality is an equality, so it’s cool to have it visualized as part of the proof.
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.
This two-point distribution is important as the distribution where Markov’s inequality is an equality, so it’s cool to have it visualized as part of the proof.
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.