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| | The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level is | | The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level is |
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| − | '''置信区间Confidence interval '''的平均''' 泊松分布Poisson distribution'''可以用泊松分布和卡方分布的累积分布函数之间的关系来表示。卡方分布本身与伽玛分布密切相关，这导致了另一种表达方式。给定一个来自平均泊松分布的观测值 k，一个带有置信水平的置信区间是 | + | 置信区间的平均泊松分布可以用泊松分布和卡方分布的累积分布函数之间的关系来表示。卡方分布本身与伽玛分布密切相关，这导致了另一种表达方式。给定一个来自平均泊松分布的观测值''k''，一个带有置信水平{{math|1 – α}}和置信区间''μ''是 |
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| − | --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布 ”改为“泊松分布 ”
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| | :<math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math> | | :<math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math> |
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| − | <math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math>
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| − | 12 chi ^ {2}(alpha/2; 2k) le mu le tfrac 12 chi ^ {2}(1-alpha/2; 2k + 2) ，</math >
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| − | or equivalently,
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| − | or equivalently,
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| | 或者等价, | | 或者等价, |
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| | :<math>F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),</math> | | :<math>F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),</math> |
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| − | <math>F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),</math>
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| − | (1-alpha/2; k，1) le mu le f ^ {-1}(1-alpha/2; k + 1,1) ，</math >
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| − | where <math>\chi^{2}(p;n)</math> is the [[quantile function]] (corresponding to a lower tail area ''p'') of the chi-squared distribution with ''n'' degrees of freedom and <math>F^{-1}(p;n,1)</math> is the quantile function of a [[gamma distribution]] with shape parameter n and scale parameter 1.{{r|Johnson2005|p=176-178|Garwood1936}} This interval is '[[exact statistics|exact]]' in the sense that its [[coverage probability]] is never less than the nominal {{math|1 – α}}.
| + | 其中<math>\chi^{2}(p;n)</math>是''n''个自由度的分位函数（对应于较低的尾部区域''p''），<math>F^{-1}(p;n,1)</math>是形状参数''n''和尺度参数1的卡方分布的分位函数。这个时间间隔是“精确的” ，因为它的覆盖概率从来没有小于名义值 {{math|1 – α}}。 |
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| − | where <math>\chi^{2}(p;n)</math> is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and <math>F^{-1}(p;n,1)</math> is the quantile function of a gamma distribution with shape parameter n and scale parameter 1. This interval is 'exact' in the sense that its coverage probability is never less than the nominal .
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| − | 其中 < math > chi ^ {2}(p; n) </math > 是 n 个自由度的分位函数，< math > f ^ {-1}(p; n，1) </math > 是形状参数 n 和尺度参数1的卡方分布的分位函数。这个时间间隔是“精确的” ，因为它的覆盖概率从来没有小于名义值。
| + | 当伽玛分布的分位数不可用时，对这个精确区间提出了精确的近似（基于 Wilson-Hilferty 变换） : |
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| − | When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the [[Chi-square distribution#Asymptotic properties|Wilson–Hilferty transformation]]):{{r|Breslow1987}}
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| − | When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation):
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| − | 当伽玛分布的分位数不可用时，对这个精确区间提出了精确的近似(基于 Wilson-Hilferty 变换) :
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| | :<math>k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3, </math> | | :<math>k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3, </math> |
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| − | <math>k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3, </math>
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| − | < math > k left (1-frac {1}{9k }-frac { z _ { alpha/2}{3 sqrt { k }右) ^ 3 le mu le (k + 1) left (1-frac {1}{9(k + 1)}} + frac { z _ { alpha/2}{3 sqrt { k + 1}右) ^ 3，</math >
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| − | where <math>z_{\alpha/2}</math> denotes the [[standard normal deviate]] with upper tail area {{math|α / 2}}.
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| − | where <math>z_{\alpha/2}</math> denotes the standard normal deviate with upper tail area .
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| − | 其中 < math > z _ { alpha/2} </math > 表示标准的正常偏差和上尾区。
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| − | For application of these formulae in the same context as above (given a sample of ''n'' measured values ''k''<sub>''i''</sub> each drawn from a Poisson distribution with mean ''λ''), one would set
| + | 其中<math>z_{\alpha/2}</math>表示标准的正常偏差和上尾区域{{math|α / 2}}。 |
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| − | For application of these formulae in the same context as above (given a sample of n measured values k<sub>i</sub> each drawn from a Poisson distribution with mean λ), one would set
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| − | 为了在与上述相同的上下文中应用这些公式(给定一个 n 个测量值 k < sub > i </sub > 每个取自一个泊松分布的平均值) ，我们将设置
| + | 为了在与上述相同的上下文中应用这些公式（给定一个''n''个测量值''k''<sub>''i''</sub>每个取自一个泊松分布的平均值''λ''） ，我们将设置 |
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| − | --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布 ”改为“泊松分布 ”
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| | :<math>k=\sum_{i=1}^n k_i ,\!</math> | | :<math>k=\sum_{i=1}^n k_i ,\!</math> |
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| − | <math>k=\sum_{i=1}^n k_i ,\!</math>
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| − | [ math > k = sum { i = 1} ^ n k _ i，
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| − | calculate an interval for ''μ'' = ''nλ'', and then derive the interval for ''λ''.
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| − | calculate an interval for μ = nλ, and then derive the interval for λ.
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| − | 计算 = n 的区间，然后推导出。
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| | + | 计算 ''μ'' = ''nλ''的区间，然后推导出''λ''。 |
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| | === 贝叶斯推理=== | | === 贝叶斯推理=== |