If U is a non-negative random variable and it has pdf $f_U(u)$, how can we prove the Markov inequality $$E[U]\geq b P(U\geq b),$$ where b is a constant?
Not sure how to prove this and haven't really gotten anywhere. Thanks for any help.
[Math] Proving the Markov inequality for a non-negative random variable
probability theoryrandom variablesstatistics
Best Answer
Comment: Sketch of proof for a continuous random variable. Can you give a justification for each step?
$$E(U) = \int_0^\infty\! xf(x)\,dx \ge \int_b^\infty\! xf(x)\,dx \ge \int_b^\infty\! bf(x)\,dx = b\int_b^\infty\! f(x)\,dx = bP(X \ge b). $$
The proof for a discrete random variable involves summations instead of integrals. Other types of integrals for a more general proof, if you know about them.