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=== Sums of Poisson-distributed random variables ===
=== Sums of Poisson-distributed random variables 泊松分布随机变量和===
: If <math>X_i \sim \operatorname{Pois}(\lambda_i)</math> for <math>i=1,\dotsc,n</math> are [[statistical independence|independent]], then <math>\sum_{i=1}^n X_i \sim \operatorname{Pois}\left(\sum_{i=1}^n \lambda_i\right)</math>.{{r|Lehmann1986|p=65}} A converse is [[Raikov's theorem]], which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.{{r|Raikov1937}}{{r|vonMises1964|p=}}
: If <math>X_i \sim \operatorname{Pois}(\lambda_i)</math> for <math>i=1,\dotsc,n</math> are [[statistical independence|independent]], then <math>\sum_{i=1}^n X_i \sim \operatorname{Pois}\left(\sum_{i=1}^n \lambda_i\right)</math>.{{r|Lehmann1986|p=65}} A converse is [[Raikov's theorem]], which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.{{r|Raikov1937}}{{r|vonMises1964|p=}}
* 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}}
* 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[['''<font color="#ff8000"> 切诺夫界Chernoff bound</font>''']]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>,
:: <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,
* 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}}
*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}}
*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>
The upper bound is proved using a standard Chernoff bound.
The upper bound is proved using a standard Chernoff bound.
利用标准的 Chernoff 界证明了上界的存在性。
利用标准的'''<font color="#ff8000"> 切诺夫界Chernoff bound</font>'''证明了上界的存在性。
* 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]].
*
*这一泊松分布是离散复合泊松分布（或断续泊松分布）在只有一个参数情况下的[[特殊情形]] 。{{r|Zhang2013|Zhang2016}}离散复合泊松分布可由一元多项式分布的极限分布导出。同时它也是[[复合泊松分布#特殊情况]] 复合泊松分布的一个特例。
* 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).
*对于足够大的值λ，（如 λ>1000)，具有均值 λ 的正态分布与变量 λ （标准差 <math>\sqrt{\lambda}</math>），是泊松分布的完美近似。如果 λ 大于10，则正态分布在适当的校正下可近似模拟，例如如果P(''X'' ≤ ''x'')，''x'' 为非负整数，则将其改为P(''X'' ≤ ''x'' + 0.5)。
:: <math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math>
:: <math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math>
* If for every ''t'' > 0 the number of arrivals in the time interval [0, ''t''] follows the Poisson distribution with mean ''λt'', then the sequence of inter-arrival times are independent and identically distributed [[exponential distribution|exponential]] random variables having mean 1/''λ''.{{r|Ross2010|p=317–319}}
* If for every ''t'' > 0 the number of arrivals in the time interval [0, ''t''] follows the Poisson distribution with mean ''λt'', then the sequence of inter-arrival times are independent and identically distributed [[exponential distribution|exponential]] random variables having mean 1/''λ''.{{r|Ross2010|p=317–319}}
*
* The [[cumulative distribution function]]s of the Poisson and [[chi-squared distribution]]s are related in the following ways:{{r|Johnson2005|p=167}}
* The [[cumulative distribution function]]s of the Poisson and [[chi-squared distribution]]s are related in the following ways:{{r|Johnson2005|p=167}}