Find an example where Markov's inequality is tight in the following sense: For each positive integer $a$, find a non-negative random variable $X$ such that $P(X\ge a)=E(X)/a$.
How to do this problem, I am really confused. Also, what is the definition of Markov's inequality?
Best Answer
Markov inequality is the integrated version of the almost sure inequality $a\mathbf 1_{X\geqslant a}\leqslant X$ hence the equality case happens if and only if $a\mathbf 1_{X\geqslant a}=X$ almost surely, that is, $X\in\{0,a\}$ almost surely.