Markov and Chebyshev are attained

ProofFoundation⏱ 15 minHigh-Dim Probabilitymarkovchebyshevtightnessextremal

Problem

For each $t>0$ and $p\in(0,1]$ construct a distribution attaining equality in Markov's inequality, and one attaining equality in Chebyshev's. Explain what these extremal distributions say about why the inequalities cannot be improved without further hypotheses.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

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Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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