Markov's inequality places an upper bound on the probability that a non-negative random variable will be greater than a fixed value. Actually, this is a special case of the more general form of the inequality: Markov is capable of placing an upper bound on the probability that a non-negative function of a random variable will be greater than a fixed value. This is very useful when we are interested in estimating the worst-case scenario that a non-negative random variable exceeds a value.
Formally, Markov's inequality states that the probability that a non-negative random variable exceeds a fixed value is less than or equal to the expectation of the random variable divided by the fixed value. If there is a non-negative function of a random variable involved, the definition changes just slightly: the probability that a non-negative function of a random variable exceeds a fixed value is less than or equal to the expectation of the non-negative function divided by the fixed value. Let's see how this upper bound works in practice, with this example taken from Wikipedia:
Imagine everyone has a (non-negative) income. Markov's inequality informs us that no more than ten percent of the population can make more than ten times the average income. The non-negative random variable in this situation is the income. The expectation of the income is just the average income. And the fixed value in this case is ten times the average income. Dividing the expectation of the income by the fixed value (ten times the expectation of the income) yields one tenth, or ten percent, as an upper bound.
In the example above, there was no non-negative function applied to the random variable. But we could apply one to the random variable, and Markov's inequality would still apply.
In practice, people often invoke Markov's inequality when they want to place an upper bound on the probability a non-negative random variable can exceed a fixed value. When showing convergence, this can be useful tool to show that the upper bound of the magnitude of the difference of two random variables is zero. Note the magnitude of the difference between two random variables is a non-negative function.