This identity is used in a simple proof of Markov's inequality.

Thus, by Markov's inequality, the probability of the first bad event above is at most .

The first moment method is a simple application of Markov's inequality for integer-valued variables.

This implies (by Markov's inequality) that is an upper bound on the probability of failure.

By Markov's Inequality, the chance that it will yield an answer before we stop it is 1/2.

The first term in comes from applying Markov's inequality to bound the probability of the first bad event (the cost is too high).

We can extend Markov's inequality to a strictly increasing and non-negative function .

This is nothing but Markov's inequality.

Now by applying Markov's inequality, we can show the decoding error probability for the first messages to be at most .

From Markov's inequality and using independence we can derive the following useful inequality:

The term Chebyshev's inequality may also refer to the Markov's inequality, especially in the context of analysis.

Since is positive, Markov's inequality holds:

The second-to-last inequality is Markov's inequality.

Common tools used in the probabilistic method include Markov's inequality, the Chernoff bound, and the Lovász local lemma.

Since in any outcome where the rounding step fails, by Markov's inequality, the conditional probability of failure is at most the conditional expectation of .

Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable.

By an application of Markov's inequality, a Las Vegas algorithm can be converted into a Monte Carlo algorithm via early termination.

In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant.

An example of an application of Markov's inequality is the fact that (assuming incomes are non-negative) no more than 1/5 of the population can have more than 5 times the average income.

It is a sharper bound than the known first or second moment based tail bounds such as Markov's inequality or Chebyshev inequality, which only yield power-law bounds on tail decay.

For any randomized trial, some variation from the mean is expected, of course, but the randomization ensures that the experimental groups have mean values that are close, due to the central limit theorem and Markov's inequality.

Observe that any Las Vegas algorithm can be converted into a Monte Carlo algorithm (via Markov's inequality), by having it output an arbitrary, possibly incorrect answer if it fails to complete within a specified time.