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第1行： − {{Probability distribution − − '''<font color="#ff8000"> 概率分布Probability distribution</font>''' + + − − '''<font color="#ff8000"> 泊松分布 Poisson distribution</font>''' − − − 类型 = 质量 − − − | pdf_image = 325px − − − | pdf_caption = The horizontal axis is the index k, the number of occurrences. λ is the expected rate of occurrences. The vertical axis is the probability of k occurrences given λ. The function is defined only at integer values of k; the connecting lines are only guides for the eye. − − | pdf _ caption = 横轴是索引 k，表示出现的次数。是预期发生率。垂直轴是给定的 k 发生概率。函数只定义在 k 的整数值上，连接线指示方向。 − − − | cdf_image = 325px − − − | cdf_caption = The horizontal axis is the index k, the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed takes on only integer values. − − | cdf _ caption = 水平轴是索引 k，表示出现的次数。因为一个'''<font color="#ff8000"> {泊松分佈 Poisson distribution</font>'''的变量只取整数值，所以 CDF 在 k 的整数和平坦的所有其他地方均不连续。 − − − | 表示法 = < math > operatorname { Pois }(lambda) </math > − − − − | support = < math > k in mathbb { n } _ 0 </math > (自然数从0开始) − − − | pdf = < math > frac { lambda ^ k e ^ {-lambda }{ k！{/math > − − − | cdf = < math > frac { Gamma (lfloor k + 1 rfloor，lambda)}{ lfloor k rfloor！{ i = 0}{ lfloor k rfloor } frac { lambda ^ i }{ i！} </math > ，或者 < math > q (lfloor k + 1 rfloor，lambda) </math > − − − (for <math>k\ge 0</math>, where <math>\Gamma(x, y)</math> is the upper incomplete gamma function, <math>\lfloor k\rfloor</math> is the floor function, and Q is the regularized gamma function) − − (对于 < math > k ge 0 </math > ，其中 < math > Gamma (x，y) </math > 是上面的不完全Γ函数，< math > lfloor k | | | | | | | | | | | | | | | | | | </math > 是下面的函数，q 是正则化的 Gamma 函数) − − − | mean = <math>\lambda</math> − − | mean = < math > > lambda </math > − − − | median = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math> − − | 中位数 = < math > > 大约1floor lambda + 1/3-0.02/lambda rfloor </math > − − − | mode = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math> − − | mode = < math > lceil lambda rceil-1，lfloor lambda rfloor − − − | variance = <math>\lambda</math> − − | variance = < math > lambda </math > − − − | skewness = <math>\lambda^{-1/2}</math> − − | skewness = < math > lambda ^ {-1/2} </math > − − − | kurtosis = <math>\lambda^{-1}</math> − − | 峭度 = < math > lambda ^ {-1} </math > − − − | entropy = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math> − − | 熵 = < math > lambda [1-log (lambda)] + e ^ {-lambda } sum _ { k = 0} ^ infty frac { lambda ^ k log (k!)}{ k！{/math > − − − (for large <math>\lambda</math>) − − (对于大的 < math > > > lambda </math >) − − <math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -{}</math><br><math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math><!--formula split with \qquad indent--> − − − 12 lambda }-frac {1}{24 lambda ^ 2}-{{{} </math > < br > < math > qfrac {19}{360 lambda ^ 3} + o left (frac {1}{1}{ lambda ^ 4}右) </math > < − − | pgf = <math>\exp[\lambda(z - 1)]</math> − − − | pgf = < math > exp [ lambda (z-1)] </math > − − | mgf = <math>\exp[\lambda (e^{t} - 1)]</math> − − − | mgf = < math > exp [ lambda (e ^ { t }-1)] </math > − − − | char = <math>\exp[\lambda (e^{it} - 1)]</math> − − | char = < math > exp [ lambda (e ^ { it }-1)] </math > − − − | fisher = <math>\frac{1}{\lambda}</math> − − | fisher = < math > frac {1}{ lambda } </math > − − }} − − − }} − −
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审校：fairywang
审校：fairywang
{{short description|Discrete probability distribution}}
{{Infobox probability distribution
| name = Poisson Distribution
| name = Poisson Distribution
| type = mass
| type = mass
| pdf_image = [[File:poisson pmf.svg|325px]]
| pdf_image = [[File:poisson pmf.svg|325px]]
| pdf_caption = The horizontal axis is the index ''k'', the number of occurrences. ''λ'' is the expected rate of occurrences. The vertical axis is the probability of ''k'' occurrences given ''λ''. The function is defined only at integer values of ''k''; the connecting lines are only guides for the eye.
| pdf_caption = The horizontal axis is the index ''k'', the number of occurrences. ''λ'' is the expected rate of occurrences. The vertical axis is the probability of ''k'' occurrences given ''λ''. The function is defined only at integer values of ''k''; the connecting lines are only guides for the eye.
| cdf_image = [[File:poisson cdf.svg|325px]]
| cdf_image = [[File:poisson cdf.svg|325px]]
| cdf_caption = The horizontal axis is the index ''k'', the number of occurrences. The CDF is discontinuous at the integers of ''k'' and flat everywhere else because a variable that is Poisson distributed takes on only integer values.
| cdf_caption = The horizontal axis is the index ''k'', the number of occurrences. The CDF is discontinuous at the integers of ''k'' and flat everywhere else because a variable that is Poisson distributed takes on only integer values.
| notation = <math>\operatorname{Pois}(\lambda)</math>
| notation = <math>\operatorname{Pois}(\lambda)</math>
| parameters = <math>\lambda\in (0, \infty) </math> (rate)
| parameters = <math>\lambda\in (0, \infty) </math> (rate)
| support = <math>k \in \mathbb{N}_0</math> ([[Natural numbers]] starting from 0)
| support = <math>k \in \mathbb{N}_0</math> ([[Natural numbers]] starting from 0)
| pdf = <math>\frac{\lambda^k e^{-\lambda}}{k!}</math>
| pdf = <math>\frac{\lambda^k e^{-\lambda}}{k!}</math>
| cdf = <math>\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}</math>, or <math>e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\ </math>, or <math>Q(\lfloor k+1\rfloor,\lambda)</math>
| cdf = <math>\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}</math>, or <math>e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\ </math>, or <math>Q(\lfloor k+1\rfloor,\lambda)</math>
(for <math>k\ge 0</math>, where <math>\Gamma(x, y)</math> is the [[upper incomplete gamma function]], <math>\lfloor k\rfloor</math> is the [[floor function]], and Q is the [[regularized gamma function]])
(for <math>k\ge 0</math>, where <math>\Gamma(x, y)</math> is the [[upper incomplete gamma function]], <math>\lfloor k\rfloor</math> is the [[floor function]], and Q is the [[regularized gamma function]])
| mean = <math>\lambda</math>
| mean = <math>\lambda</math>
| median = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math>
| median = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math>
| mode = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math>
| mode = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math>
| variance = <math>\lambda</math>
| variance = <math>\lambda</math>
| skewness = <math>\lambda^{-1/2}</math>
| skewness = <math>\lambda^{-1/2}</math>
| kurtosis = <math>\lambda^{-1}</math>
| kurtosis = <math>\lambda^{-1}</math>
| entropy = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math>
| entropy = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math>
(for large <math>\lambda</math>)
(for large <math>\lambda</math>)
<math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -{}</math><br><math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math><!--formula split with \qquad indent-->
<math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -{}</math><br><math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math><!--formula split with \qquad indent-->
| pgf = <math>\exp[\lambda(z - 1)]</math>
| pgf = <math>\exp[\lambda(z - 1)]</math>
| mgf = <math>\exp[\lambda (e^{t} - 1)]</math>
| mgf = <math>\exp[\lambda (e^{t} - 1)]</math>
| char = <math>\exp[\lambda (e^{it} - 1)]</math>
| char = <math>\exp[\lambda (e^{it} - 1)]</math>
| fisher = <math>\frac{1}{\lambda}</math>
| fisher = <math>\frac{1}{\lambda}</math>
}}
}}
In [[probability theory]] and [[statistics]], the '''Poisson distribution''' ({{IPAc-en|'|p|w|ɑː|s|ɒ|n}}; {{IPA-fr|pwasɔ̃}}), named after [[France|French]] mathematician [[Siméon Denis Poisson]], is a [[discrete probability distribution]] that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and [[Statistical independence|independently]] of the time since the last event.{{r|Haight1967}} The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
In [[probability theory]] and [[statistics]], the '''Poisson distribution''' ({{IPAc-en|'|p|w|ɑː|s|ɒ|n}}; {{IPA-fr|pwasɔ̃}}), named after [[France|French]] mathematician [[Siméon Denis Poisson]], is a [[discrete probability distribution]] that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and [[Statistical independence|independently]] of the time since the last event.{{r|Haight1967}} The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.