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第240行： − + − − * The Poisson distributions are [[Infinite divisibility (probability)|infinitely divisible]] probability distributions.{{r|Laha1979|p=233}}{{r|Johnson2005|p=164}} − − * The directed [[Kullback–Leibler divergence]] of <math>\operatorname{Pois}(\lambda_0)</math> from <math>\operatorname{Pois}(\lambda)</math> is given by − + − <math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math> − + − * Bounds for the tail probabilities of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> can be derived using a [[Chernoff bound]] argument.{{r|Mitzenmacher2005|p=97-98}} − *泊松随机变量尾概率的界<math> X \sim \operatorname{Pois}(\lambda)</math> 可以用[[''' 切诺夫界Chernoff bound''']]参数派生{{r|Mitzenmacher2005|p=97-98}} − − <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>, − − (x x x) leq frac {(e lambda) ^ x e ^ {-lambda }{ x ^ x } ，text { for } x > lambda </math > , − − <math> P(X \leq x) \leq \frac{(e \lambda)^x e^{-\lambda} }{x^x}, \text{ for } x < \lambda.</math> − − < math > p (x leq x) leq frac {(e lambda) ^ x e ^ {-lambda }{ x ^ x } ，text { for } x < lambda. </math > + − − * The upper tail probability can be tightened (by a factor of at least two) as follows:{{r|Short2013}} − − *长尾概率可被收紧（至少两倍）如下：{{r|Short2013}} − <math> P(X \geq x) \leq \frac{e^{-\operatorname{D}_{\text{KL}}(x\mid\lambda)}}{\max{(2, \sqrt{4\pi\operatorname{D}_{\text{KL}}(x\mid\lambda)}})}, \text{ for } x > \lambda,</math> − − < math > p (x geq x) leq frac { e ^ {-operatorname { d }{ text { KL }(x mid lambda)}}{ max {(2，sqrt {4 pi operatorname { d }{ text { KL }(x mid lambda)}}}}) ，text { for } x > lambda，</math > − − − : where <math>\operatorname{D}_{\text{KL}}(x\mid\lambda)</math> is the directed Kullback–Leibler divergence, as described above. − where <math>\operatorname{D}_{\text{KL}}(x\mid\lambda)</math> is the directed Kullback–Leibler divergence, as described above.+ − 其中 < math > operatorname { d } _ { text { KL }}(x mid lambda) </math > 是指向的 Kullback-Leibler 分歧，如上所述。 − + − − * Inequalities that relate the distribution function of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> to the [[Standard normal distribution]] function <math> \Phi(x) </math> are as follows:{{r|Short2013}} − − − <math> \Phi\left(\operatorname{sign}(k-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(k\mid\lambda)}\right) < P(X \leq k) < \Phi\left(\operatorname{sign}(k-\lambda+1)\sqrt{2\operatorname{D}_{\text{KL}}(k+1\mid\lambda)}\right), \text{ for } k > 0,</math> − − < math > Phi left (operatorname { sign }(k-lambda) sqrt {2 operatorname { d }{ text { KL }(k mid lambda)}右) < p (x leq k) < left (operatorname { sign }(k-lambda + 1) sqrt {2 operatorname { d }{ text { KL }}}(k + 1 mid lambda)}右) ，text { for } k > 0，</math > − − − − : where <math>\operatorname{D}_{\text{KL}}(k\mid\lambda)</math> is again the directed Kullback–Leibler divergence. − − where <math>\operatorname{D}_{\text{KL}}(k\mid\lambda)</math> is again the directed Kullback–Leibler divergence. − − 其中 < math > 操作者名{ d } _ { text { KL }(k mid lambda) </math > 仍然是有向的 Kullback-Leibler 分歧。 + +
→Other properties 其他特性
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=== Other properties 其他特性===
=== 其他特性===
*泊松分布是[[无限可除性（概率）|无限可除]] 概率分布，{{r|Laha1979|p=233}}{{r|Johnson2005|p=164}}。
*泊松分布是[[无限可除性（概率）|无限可除]] 概率分布，{{r|Laha1979|p=233}}{{r|Johnson2005|p=164}}。
*<math>\operatorname{Pois}(\lambda_0)</math> from <math>\operatorname{Pois}(\lambda)</math> 的直接''' 相对熵（K-L散度）[[Kullback–Leibler divergence]]'''由以下给出：
*<math>\operatorname{Pois}(\lambda_0)</math> from <math>\operatorname{Pois}(\lambda)</math> 的直接'''相对熵（K-L散度）[[Kullback–Leibler divergence]]'''由以下给出：
:: <math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math>
:: <math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math>
< math > operatorname { d }{ text { KL }}(lambda mid lambda _ 0) = lambda _ 0-lambda + lambda log frac { lambda }{ lambda _ 0} . </math >
*泊松随机变量尾概率的界<math> X \sim \operatorname{Pois}(\lambda)</math> 可以用[[切诺夫界 Chernoff bound]]参数派生
:: <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,
:: <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,
:: <math> P(X \leq x) \leq \frac{(e \lambda)^x e^{-\lambda} }{x^x}, \text{ for } x < \lambda.</math>
:: <math> P(X \leq x) \leq \frac{(e \lambda)^x e^{-\lambda} }{x^x}, \text{ for } x < \lambda.</math>
*长尾概率可被收紧（至少两倍）如下：
:: <math> P(X \geq x) \leq \frac{e^{-\operatorname{D}_{\text{KL}}(x\mid\lambda)}}{\max{(2, \sqrt{4\pi\operatorname{D}_{\text{KL}}(x\mid\lambda)}})}, \text{ for } x > \lambda,</math>
:: <math> P(X \geq x) \leq \frac{e^{-\operatorname{D}_{\text{KL}}(x\mid\lambda)}}{\max{(2, \sqrt{4\pi\operatorname{D}_{\text{KL}}(x\mid\lambda)}})}, \text{ for } x > \lambda,</math>
其中<math>\operatorname{D}_{\text{KL}}(x\mid\lambda)</math>是指向的 Kullback-Leibler 分歧，如上所述。
*关于泊松随机变量分布函数的不等式 <math> X \sim \operatorname{Pois}(\lambda)</math>与 标准正态分布函数<math> \Phi(x) </math> 如下:
*关于泊松随机变量分布函数的不等式 <math> X \sim \operatorname{Pois}(\lambda)</math>与 标准正态分布函数<math> \Phi(x) </math> 如下:{{r|Short2013}}
:: <math> \Phi\left(\operatorname{sign}(k-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(k\mid\lambda)}\right) < P(X \leq k) < \Phi\left(\operatorname{sign}(k-\lambda+1)\sqrt{2\operatorname{D}_{\text{KL}}(k+1\mid\lambda)}\right), \text{ for } k > 0,</math>
:: <math> \Phi\left(\operatorname{sign}(k-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(k\mid\lambda)}\right) < P(X \leq k) < \Phi\left(\operatorname{sign}(k-\lambda+1)\sqrt{2\operatorname{D}_{\text{KL}}(k+1\mid\lambda)}\right), \text{ for } k > 0,</math>
其中 <math>\operatorname{D}_{\text{KL}}(k\mid\lambda)</math>仍然是有向的 Kullback-Leibler 分歧。
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=== Poisson races 泊松族群===
=== Poisson races 泊松族群===