马尔可夫不等式 Markov's inequality

If X is any nonnegative integrable random variable and a > 0, then

In the language of measure theory, Markov's inequality states that if (X, Σ, μ) is a measure space, ƒ is a measurable extended real-valued function, and , then

证明

For any event E, let I_E be the indicator random variable of E, that is, I_E = 1 if E occurs and I_E = 0 otherwise.

Using this notation, we have I_(X ≥ a) = 1 if the event X ≥ a occurs, and I_(X ≥ a) = 0 if X < a. Then, given a > 0,

which is clear if we consider the two possible values of I_(X ≥ a). If X < a, then I_(X ≥ a) = 0, and so aI_(X ≥ a) = 0 ≤ X. Otherwise, we have X ≥ a, for which I_(X ≥ a) = 1 and so aI_(X ≥ a) = a ≤ X.

Since  is a monotonically increasing function, taking expectation of both sides of an inequality cannot reverse it. Therefore

Now, using linearity of expectations, the left side of this inequality is the same as

Thus we have

and since a > 0, we can divide both sides by a.