马尔可夫不等式给出了一个实值随机变量取值大于等于某个特定数值的概率的上限。设X是一个随机变量，a>0为正实数，那么以下不等式成立[ 1]
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{\displaystyle \mathbb {P} (|X|\geq a)\leq {\frac {\mathbb {E} (|X|)}{a}}.}
这个不等式可以推广。对所有的单调严格递增的非零函数
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{\displaystyle \Phi }
[ 1]
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{\displaystyle \mathbb {P} (X\geq a)=\mathbb {P} (\Phi (X)\geq \Phi (a))\leq {\frac {\mathbb {E} (\Phi (X))}{\Phi (a)}}.}
霍夫丁不等式适用于有界的随机变量。设有两两独立的一系列随机变量
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{\displaystyle X_{1},\dots ,X_{n}\!}
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{\displaystyle 1\leq i\leq n}
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{\displaystyle X_{i}}
几乎 有界的变量，即满足：
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{\displaystyle \mathbb {P} (X_{i}\in [a_{i},b_{i}])=1.\!}
那么这n个随机变量的经验期望：
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{\displaystyle {\overline {X}}={\frac {X_{1}+\cdots +X_{n}}{n}}}
满足以下的不等式[ 1] [ 2]
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{\displaystyle \mathbb {P} ({\overline {X}}-\mathbb {E} [{\overline {X}}]\geq t)\leq \exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!}
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{\displaystyle \mathbb {P} (|{\overline {X}}-\mathbb {E} [{\overline {X}}]|\geq t)\leq 2\exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!}
Efron–Stein不等式给出了随机变量方差的一个上限估计。设有两两独立的随机变量
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{\displaystyle X_{1}\dots X_{n}}
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{\displaystyle X_{1}'\dots X_{n}'}
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{\displaystyle X_{i}'}
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{\displaystyle X_{i}}
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{\displaystyle X=(X_{1},\dots ,X_{n}),X^{(i)}=(X_{1},\dots ,X_{i-1},X_{i}',X_{i+1},\dots ,X_{n})}
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{\displaystyle \mathrm {Var} (f(X))\leq {\frac {1}{2}}\sum _{i=1}^{n}E[(f(X)-f(X^{(i)}))^{2}].}
[ 1]
^ 1.0 1.1 1.2 1.3 1.4 Stéphane Boucheron, Gabor Lugosi, Olivier Bousquet. Concentration Inequalities (PDF) . Université de Paris-Sud, Laboratoire d'Informatique. [2012年9月7日] . （原始内容存档 (PDF) 于2020年9月28日）. （英文） ^ Wassily Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 (301): 13–30, March 1963. (JSTOR )（英文）
![{\displaystyle \operatorname {Var} (X)=\mathbb {E} [(X-\mathbb {E} (X))^{2}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/903a713a326dea3725798b9f1d01eb0b2f82270f)
![{\displaystyle \mathbb {P} (X_{i}\in [a_{i},b_{i}])=1.\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6f13669eacf3e303df1837041e91ce704f7345)
![{\displaystyle \mathbb {P} ({\overline {X}}-\mathbb {E} [{\overline {X}}]\geq t)\leq \exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2384a4ef64dbf81cdfa5fe6f517ef35494a6b022)
![{\displaystyle \mathbb {P} (|{\overline {X}}-\mathbb {E} [{\overline {X}}]|\geq t)\leq 2\exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51f4eeb59eab8345db944e6f7a2c91fd8a6146c4)
![{\displaystyle \mathrm {Var} (f(X))\leq {\frac {1}{2}}\sum _{i=1}^{n}E[(f(X)-f(X^{(i)}))^{2}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c25450c4c542c8a1dd444386d89095d03ae96e97)