If X, then formula_27, and so formula_32. Otherwise, we have formula_26, for which formula_34 and so formula_35. Since formula_36 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". We may assume that the function formula_40 is non-negative, since only its absolute value enters in the equation. Now, consider the real-valued function "s" on "X" given by  Then formula_42. By the definition of the Lebesgue integral  and since formula_44, both sides can be divided by formula_45, obtaining  Chebyshev's inequality uses the variance to bound the probability that a random variable deviates far from the mean. Specifically,   for any . Here is the variance of X, defined as:  Chebyshev's inequality follows from Markov's inequality by considering the random variable  and the constant formula_50 for which Markov's inequality reads  This argument can be summarized (where "MI" indicates use of Markov's inequality):   Assuming no income is negative, Markov's inequality shows that no more than 1/5 of the population can have more than 5 times the average income.