区间估计

我就是我 2023-06-27 03:40 43阅读 0赞

<table> 
 <thead> 
  <tr> 
   <th>求谁</th> 
   <th><span><span><span> σ 是 否 已 知 \sigma是否已知 </span><span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span style="margin-right: 0.03588em;">σ</span><span>是</span><span>否</span><span>已</span><span>知</span></span></span></span></span></th> 
   <th>置信区间</th> 
   <th>置信区间</th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td>求<span><span><span> μ \mu </span><span><span><span style="height: 0.625em; vertical-align: -0.19444em;"></span><span>μ</span></span></span></span></span></td> 
   <td><span><span><span> σ 未 知 \sigma未知 </span><span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span style="margin-right: 0.03588em;">σ</span><span>未</span><span>知</span></span></span></span></span></td> 
   <td><span><span><span> 1 − α 1-\alpha </span><span><span><span style="height: 0.72777em; vertical-align: -0.08333em;"></span><span>1</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span style="margin-right: 0.0037em;">α</span></span></span></span></span></td> 
   <td><span><span><span> ( X ‾ − σ ( n ) Z α 2 , X ‾ + σ ( n ) Z α 2 ) (\overline X- \frac{\sigma}{\sqrt(n)}Z_{\frac{\alpha}{2}},\overline X+\frac{\sigma}{\sqrt(n)}Z_{\frac{\alpha}{2}}) </span><span><span><span style="height: 1.13333em; vertical-align: -0.25em;"></span><span>(</span><span><span><span><span style="height: 0.88333em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.07847em;">X</span></span><span style="top: -3.80333em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span></span></span></span></span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1.71293em; vertical-align: -0.8296em;"></span><span><span></span><span><span><span><span style="height: 0.695392em;"><span style="top: -2.46416em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 1.03691em;"><span style="top: -3.42857em;"><span style="height: 3.42857em;"></span><span style="padding-left: 1.19em;">(</span></span><span style="top: -3.00891em;"><span style="height: 3.42857em;"></span><span style="min-width: 0.853em; height: 1.54286em;">   </span></span></span><span></span></span><span><span style="height: 0.419661em;"><span></span></span></span></span></span><span>n</span><span>)</span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.394em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.03588em;">σ</span></span></span></span></span><span></span></span><span><span style="height: 0.8296em;"><span></span></span></span></span></span><span></span></span><span><span style="margin-right: 0.07153em;">Z</span><span><span><span><span style="height: 0.33408em;"><span style="top: -2.85em; margin-left: -0.07153em; margin-right: 0.05em;"><span style="height: 3em;"></span><span><span><span><span></span><span><span><span><span style="height: 0.691543em;"><span style="top: -2.656em;"><span style="height: 3em;"></span><span><span><span>2</span></span></span></span><span style="top: -3.2255em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span><span style="top: -3.384em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.0037em;">α</span></span></span></span></span><span></span></span><span><span style="height: 0.344em;"><span></span></span></span></span></span><span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.3908em;"><span></span></span></span></span></span></span><span>,</span><span style="margin-right: 0.166667em;"></span><span><span><span><span style="height: 0.88333em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.07847em;">X</span></span><span style="top: -3.80333em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span></span></span></span></span><span style="margin-right: 0.222222em;"></span><span>+</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1.5796em; vertical-align: -0.8296em;"></span><span><span></span><span><span><span><span style="height: 0.695392em;"><span style="top: -2.46416em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 1.03691em;"><span style="top: -3.42857em;"><span style="height: 3.42857em;"></span><span style="padding-left: 1.19em;">(</span></span><span style="top: -3.00891em;"><span style="height: 3.42857em;"></span><span style="min-width: 0.853em; height: 1.54286em;">   </span></span></span><span></span></span><span><span style="height: 0.419661em;"><span></span></span></span></span></span><span>n</span><span>)</span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.394em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.03588em;">σ</span></span></span></span></span><span></span></span><span><span style="height: 0.8296em;"><span></span></span></span></span></span><span></span></span><span><span style="margin-right: 0.07153em;">Z</span><span><span><span><span style="height: 0.33408em;"><span style="top: -2.85em; margin-left: -0.07153em; margin-right: 0.05em;"><span style="height: 3em;"></span><span><span><span><span></span><span><span><span><span style="height: 0.691543em;"><span style="top: -2.656em;"><span style="height: 3em;"></span><span><span><span>2</span></span></span></span><span style="top: -3.2255em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span><span style="top: -3.384em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.0037em;">α</span></span></span></span></span><span></span></span><span><span style="height: 0.344em;"><span></span></span></span></span></span><span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.3908em;"><span></span></span></span></span></span></span><span>)</span></span></span></span></span></td> 
  </tr> 
  <tr> 
   <td>求<span><span><span> μ \mu </span><span><span><span style="height: 0.625em; vertical-align: -0.19444em;"></span><span>μ</span></span></span></span></span></td> 
   <td><span><span><span> σ 已 知 \sigma已知 </span><span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span style="margin-right: 0.03588em;">σ</span><span>已</span><span>知</span></span></span></span></span></td> 
   <td><span><span><span> 1 − α 1-\alpha </span><span><span><span style="height: 0.72777em; vertical-align: -0.08333em;"></span><span>1</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span style="margin-right: 0.0037em;">α</span></span></span></span></span></td> 
   <td><span><span><span> ( X ‾ − σ ( n ) t α 2 ( n − 1 ) , X ‾ + σ ( n ) t α 2 ( n − 1 ) ) (\overline X- \frac{\sigma}{\sqrt(n)}t_{\frac{\alpha}{2}}(n-1),\overline X+\frac{\sigma}{\sqrt(n)}t_{\frac{\alpha}{2}}(n-1)) </span><span><span><span style="height: 1.13333em; vertical-align: -0.25em;"></span><span>(</span><span><span><span><span style="height: 0.88333em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.07847em;">X</span></span><span style="top: -3.80333em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span></span></span></span></span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1.5796em; vertical-align: -0.8296em;"></span><span><span></span><span><span><span><span style="height: 0.695392em;"><span style="top: -2.46416em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 1.03691em;"><span style="top: -3.42857em;"><span style="height: 3.42857em;"></span><span style="padding-left: 1.19em;">(</span></span><span style="top: -3.00891em;"><span style="height: 3.42857em;"></span><span style="min-width: 0.853em; height: 1.54286em;">   </span></span></span><span></span></span><span><span style="height: 0.419661em;"><span></span></span></span></span></span><span>n</span><span>)</span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.394em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.03588em;">σ</span></span></span></span></span><span></span></span><span><span style="height: 0.8296em;"><span></span></span></span></span></span><span></span></span><span><span>t</span><span><span><span><span style="height: 0.33408em;"><span style="top: -2.85em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 3em;"></span><span><span><span><span></span><span><span><span><span style="height: 0.691543em;"><span style="top: -2.656em;"><span style="height: 3em;"></span><span><span><span>2</span></span></span></span><span style="top: -3.2255em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span><span style="top: -3.384em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.0037em;">α</span></span></span></span></span><span></span></span><span><span style="height: 0.344em;"><span></span></span></span></span></span><span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.3908em;"><span></span></span></span></span></span></span><span>(</span><span>n</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1.13333em; vertical-align: -0.25em;"></span><span>1</span><span>)</span><span>,</span><span style="margin-right: 0.166667em;"></span><span><span><span><span style="height: 0.88333em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.07847em;">X</span></span><span style="top: -3.80333em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span></span></span></span></span><span style="margin-right: 0.222222em;"></span><span>+</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1.5796em; vertical-align: -0.8296em;"></span><span><span></span><span><span><span><span style="height: 0.695392em;"><span style="top: -2.46416em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 1.03691em;"><span style="top: -3.42857em;"><span style="height: 3.42857em;"></span><span style="padding-left: 1.19em;">(</span></span><span style="top: -3.00891em;"><span style="height: 3.42857em;"></span><span style="min-width: 0.853em; height: 1.54286em;">   </span></span></span><span></span></span><span><span style="height: 0.419661em;"><span></span></span></span></span></span><span>n</span><span>)</span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.394em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.03588em;">σ</span></span></span></span></span><span></span></span><span><span style="height: 0.8296em;"><span></span></span></span></span></span><span></span></span><span><span>t</span><span><span><span><span style="height: 0.33408em;"><span style="top: -2.85em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 3em;"></span><span><span><span><span></span><span><span><span><span style="height: 0.691543em;"><span style="top: -2.656em;"><span style="height: 3em;"></span><span><span><span>2</span></span></span></span><span style="top: -3.2255em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span><span style="top: -3.384em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.0037em;">α</span></span></span></span></span><span></span></span><span><span style="height: 0.344em;"><span></span></span></span></span></span><span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.3908em;"><span></span></span></span></span></span></span><span>(</span><span>n</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1em; vertical-align: -0.25em;"></span><span>1</span><span>)</span><span>)</span></span></span></span></span></td> 
  </tr> 
  <tr> 
   <td><span><span><span> 求 σ 求\sigma </span><span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span>求</span><span style="margin-right: 0.03588em;">σ</span></span></span></span></span></td> 
   <td><span><span><span> σ 未 知 \sigma未知 </span><span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span style="margin-right: 0.03588em;">σ</span><span>未</span><span>知</span></span></span></span></span></td> 
   <td><span><span><span> 1 − α 1-\alpha </span><span><span><span style="height: 0.72777em; vertical-align: -0.08333em;"></span><span>1</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 0.43056em; vertical-align: 0em;"></span><span style="margin-right: 0.0037em;">α</span></span></span></span></span></td> 
   <td><span><span><span> ( X ‾ − n − 1 S χ α 2 2 ( n − 1 ) , X ‾ + n − 1 S χ 1 − α 2 2 ( n − 1 ) ) (\overline X - \frac{\sqrt{n-1}S}{\sqrt{\chi^2_{\frac{\alpha}{2}}(n-1)}},\overline X + \frac{\sqrt{n-1}S}{\sqrt{\chi^2_{1-\frac{\alpha}{2}}(n-1)}}) </span><span><span><span style="height: 1.13333em; vertical-align: -0.25em;"></span><span>(</span><span><span><span><span style="height: 0.88333em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.07847em;">X</span></span><span style="top: -3.80333em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span></span></span></span></span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 2.6676em; vertical-align: -1.6296em;"></span><span><span></span><span><span><span><span style="height: 1.038em;"><span style="top: -2.24268em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 1.35331em;"><span style="top: -4.57143em;"><span style="height: 4.57143em;"></span><span style="padding-left: 1.42857em;"><span><span>χ</span><span><span><span><span style="height: 0.805114em;"><span style="top: -2.56483em; margin-left: 0em; margin-right: 0.0714286em;"><span style="height: 3em;"></span><span><span><span><span></span><span><span><span><span style="height: 0.81756em;"><span style="top: -2.468em;"><span style="height: 3em;"></span><span><span>2</span></span></span><span style="top: -3.2255em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span><span style="top: -3.387em;"><span style="height: 3em;"></span><span><span style="margin-right: 0.0037em;">α</span></span></span></span><span></span></span><span><span style="height: 0.532em;"><span></span></span></span></span></span><span></span></span></span></span></span><span style="top: -3.3448em; margin-right: 0.0714286em;"><span style="height: 3em;"></span><span><span>2</span></span></span></span><span></span></span><span><span style="height: 0.815171em;"><span></span></span></span></span></span></span><span>(</span><span>n</span><span>−</span><span>1</span><span>)</span></span></span><span style="top: -3.32531em;"><span style="height: 4.57143em;"></span><span style="min-width: 1.02em; height: 2.68571em;">   </span></span></span><span></span></span><span><span style="height: 1.24612em;"><span></span></span></span></span></span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.42817em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 0.87118em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="padding-left: 0.833em;"><span>n</span><span>−</span><span>1</span></span></span><span style="top: -2.83118em;"><span style="height: 3em;"></span><span style="min-width: 0.853em; height: 1.08em;">   </span></span></span><span></span></span><span><span style="height: 0.16882em;"><span></span></span></span></span></span><span style="margin-right: 0.05764em;">S</span></span></span></span></span><span></span></span><span><span style="height: 1.6296em;"><span></span></span></span></span></span><span></span></span><span>,</span><span style="margin-right: 0.166667em;"></span><span><span><span><span style="height: 0.88333em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.07847em;">X</span></span><span style="top: -3.80333em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span></span></span></span></span><span style="margin-right: 0.222222em;"></span><span>+</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 2.6676em; vertical-align: -1.6296em;"></span><span><span></span><span><span><span><span style="height: 1.038em;"><span style="top: -2.24268em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 1.35331em;"><span style="top: -4.57143em;"><span style="height: 4.57143em;"></span><span style="padding-left: 1.42857em;"><span><span>χ</span><span><span><span><span style="height: 0.805114em;"><span style="top: -2.56483em; margin-left: 0em; margin-right: 0.0714286em;"><span style="height: 3em;"></span><span><span><span>1</span><span>−</span><span><span></span><span><span><span><span style="height: 0.81756em;"><span style="top: -2.468em;"><span style="height: 3em;"></span><span><span>2</span></span></span><span style="top: -3.2255em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span><span style="top: -3.387em;"><span style="height: 3em;"></span><span><span style="margin-right: 0.0037em;">α</span></span></span></span><span></span></span><span><span style="height: 0.532em;"><span></span></span></span></span></span><span></span></span></span></span></span><span style="top: -3.3448em; margin-right: 0.0714286em;"><span style="height: 3em;"></span><span><span>2</span></span></span></span><span></span></span><span><span style="height: 0.815171em;"><span></span></span></span></span></span></span><span>(</span><span>n</span><span>−</span><span>1</span><span>)</span></span></span><span style="top: -3.32531em;"><span style="height: 4.57143em;"></span><span style="min-width: 1.02em; height: 2.68571em;">   </span></span></span><span></span></span><span><span style="height: 1.24612em;"><span></span></span></span></span></span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.42817em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 0.87118em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="padding-left: 0.833em;"><span>n</span><span>−</span><span>1</span></span></span><span style="top: -2.83118em;"><span style="height: 3em;"></span><span style="min-width: 0.853em; height: 1.08em;">   </span></span></span><span></span></span><span><span style="height: 0.16882em;"><span></span></span></span></span></span><span style="margin-right: 0.05764em;">S</span></span></span></span></span><span></span></span><span><span style="height: 1.6296em;"><span></span></span></span></span></span><span></span></span><span>)</span></span></span></span></span></td> 
  </tr> 
 </tbody> 
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约定不等于承诺〃/ 2022年07月14日 14:57/ 0 赞/ 229 阅读

相关非参数估计：核密度估计KDE

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傷城~/ 2022年07月14日 14:57/ 0 赞/ 242 阅读

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这里的场景是：给出一组数据，估计一个区间，使得区间内的数据大致成均匀分布，并且保证区间内的数据能够包含总体的大部分数据。（统计学中“区间估计”指的是，用来描述根据样本求得

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