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| * The upper tail probability can be tightened (by a factor of at least two) as follows:{{r|Short2013}} | | * The upper tail probability can be tightened (by a factor of at least two) as follows:{{r|Short2013}} |
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| + | *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> |
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| < 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 > |
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| * 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}} | | * 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}} |
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− | | + | *关于泊松随机变量分布函数的不等式 <math> X \sim \operatorname{Pois}(\lambda)</math>对 标准正态分布函数<math> \Phi(x) </math> are as follows:{{r|Short2013}} |
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| :: <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> |
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− | === Poisson races === | + | === Poisson races 泊松族群=== |
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− | == Related distributions == | + | == '''<font color="#ff8000"> 相关分布'Related distributions</font>''== |
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− | ===General=== | + | ===Genera通常l=== |
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| * If <math>X_1 \sim \mathrm{Pois}(\lambda_1)\,</math> and <math>X_2 \sim \mathrm{Pois}(\lambda_2)\,</math> are independent, then the difference <math> Y = X_1 - X_2</math> follows a [[Skellam distribution]]. | | * If <math>X_1 \sim \mathrm{Pois}(\lambda_1)\,</math> and <math>X_2 \sim \mathrm{Pois}(\lambda_2)\,</math> are independent, then the difference <math> Y = X_1 - X_2</math> follows a [[Skellam distribution]]. |
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| * The Poisson distribution is a [[special case]] of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter.{{r|Zhang2013|Zhang2016}} The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a [[compound Poisson distribution#Special cases|special case]] of a [[compound Poisson distribution]]. | | * The Poisson distribution is a [[special case]] of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter.{{r|Zhang2013|Zhang2016}} The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a [[compound Poisson distribution#Special cases|special case]] of a [[compound Poisson distribution]]. |
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| * For sufficiently large values of λ, (say λ>1000), the [[normal distribution]] with mean λ and variance λ (standard deviation <math>\sqrt{\lambda}</math>) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [[continuity correction]] is performed, i.e., if P(''X'' ≤ ''x''), where ''x'' is a non-negative integer, is replaced by P(''X'' ≤ ''x'' + 0.5). | | * For sufficiently large values of λ, (say λ>1000), the [[normal distribution]] with mean λ and variance λ (standard deviation <math>\sqrt{\lambda}</math>) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [[continuity correction]] is performed, i.e., if P(''X'' ≤ ''x''), where ''x'' is a non-negative integer, is replaced by P(''X'' ≤ ''x'' + 0.5). |
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