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Markov's Inequality

If x takes only nonnegative values, then

P(x>=a)<=(<x>)/a.
(1)

To prove the theorem, write

<x>=int_0^inftyxP(x)dx
(2)
=int_0^axP(x)dx+int_a^inftyxP(x)dx.
(3)

Since P(x) is a probability density, it must be >=0. We have stipulated that x>=0, so

<x>=int_0^axP(x)dx+int_a^inftyxP(x)dx
(4)
>=int_a^inftyxP(x)dx
(5)
>=int_a^inftyaP(x)dx
(6)
=aint_a^inftyP(x)dx
(7)
=aP(x>=a).
(8)

Q.E.D.

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Cite this as:

Weisstein, Eric W. "Markov's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MarkovsInequality.html

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