Concept in probability theory
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality.
It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Bienaymé's inequality.
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. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.
Statement.
If X is a nonnegative random variable and "a"  ; 0, then the probability
that X is at least a is at most the expectation of X divided by a:
formula_1
When formula_2, we can take formula_3 for formula_4 to rewrite the previous inequality as
formula_5
In the language of measure theory, Markov's inequality states that if ("X", Σ, "μ") is a measure space, formula_6 is a measurable extended real-valued function, and "ε" ; 0, then
formula_7
This measure-theoretic definition is sometimes referred to as Chebyshev's inequality.
Extended version for nondecreasing functions.
If φ is a nondecreasing nonnegative function, X is a (not necessarily nonnegative) random variable, and "φ"("a") ; 0, then
formula_8
An immediate corollary, using higher moments of X supported on values larger than 0, is
formula_9
The uniformly randomized Markov's inequality.
If X is a nonnegative random variable and "a"  ; 0, and U is a uniformly distributed random variable on formula_10 that is independent of X, then
formula_11
Since U is almost surely smaller than one, this bound is strictly stronger than Markov's inequality. Remarkably, U cannot be replaced by any constant smaller than one, meaning that deterministic improvements to Markov's inequality cannot exist in general. While Markov's inequality holds with equality for distributions supported on formula_12, the above randomized variant holds with equality for any distribution that is bounded on formula_13.
Proofs.
We separate the case in which the measure space is a probability space from the more general case because the probability case is more accessible for the general reader.
Intuition.
formula_14 is non-negative and formula_15 is larger than or equal to formula_16 because the conditional expectation only takes into account of values larger than or equal to formula_16 which r.v. formula_18 can take.
Property 1: formula_19
Given a non-negative random variable formula_18, the conditional expectation formula_21 because formula_22. Also, probabilities are always non-negative, i.e., formula_23. Thus, the product:
formula_19.
This is intuitive since conditioning on formula_25 still results in non-negative values, ensuring the product remains non-negative.
Property 2: formula_26
For formula_27, the expected value given formula_28 is at least formula_29. Multiplying both sides by formula_30, we get:
formula_31.
This is intuitive since all values considered are at least formula_16, making their average also greater than or equal to formula_16.
Hence intuitively, formula_34, which directly leads to formula_35.
Probability-theoretic proof.
Method 1:
From the definition of expectation: 
formula_36
However, X is a non-negative random variable thus, 
formula_37
From this we can derive, 
formula_38
From here, dividing through by formula_16 allows us to see that 
formula_40
Method 2:
For any event formula_41, let formula_42 be the indicator random variable of formula_43, that is, formula_44 if formula_41 occurs and formula_46 otherwise.
Using this notation, we have formula_47 if the event formula_48 occurs, and formula_49 if formula_50,
formula_51
which is clear if we consider the two possible values of formula_48. If ;math ;X, then formula_49, and so formula_54. Otherwise, we have formula_48, for which formula_56 and so formula_57.
Since formula_58 is a monotonically increasing function, taking expectation of both sides of an inequality cannot reverse it. Therefore,
formula_59
Now, using linearity of expectations, the left side of this inequality is the same as
formula_60
Thus we have
formula_61
and since "a"  ; 0, we can divide both sides by "a".
Measure-theoretic proof.
We may assume that the function formula_6 is non-negative, since only its absolute value enters in the equation. Now, consider the real-valued function "s" on "X" given by
formula_63
Then formula_64. By the definition of the Lebesgue integral
formula_65
and since formula_66, both sides can be divided by formula_67, obtaining
formula_68
Discrete case.
We now provide a proof for the special case when formula_18 is a discrete random variable which only takes on non-negative integer values. 
Let formula_16 be a positive integer. By definition formula_71 
formula_72 
formula_73 
formula_74
formula_75
formula_76
Dividing by formula_16 yields the desired result.
Corollaries.
Chebyshev's inequality.
Chebyshev's inequality uses the variance to bound the probability that a random variable deviates far from the mean. Specifically, 
formula_78
for any "a" ; 0. Here Var("X") is the variance of X, defined as:
formula_79
Chebyshev's inequality follows from Markov's inequality by considering the random variable
 formula_80
and the constant formula_81 for which Markov's inequality reads
 formula_82
This argument can be summarized (where "MI" indicates use of Markov's inequality):
formula_83
Examples.
Assuming no income is negative, Markov's inequality shows that no more than 10% (1/10) of the population can have more than 10 times the average income. 
Another simple example is as follows: Andrew makes 4 mistakes on average on his Statistics course tests. The best upper bound on the probability that Andrew will do at least 10 mistakes is 0.4 as formula_90 Note that Andrew might do exactly 10 mistakes with probability 0.4 and make no mistakes with probability 0.6; the expectation is exactly 4 mistakes.
References.
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