标签：inf nbsp src sum png ima 技术 图片 sig
$$ E\left[ \sigma _{MLE}^{2} \right] =E\left[ \frac{1}{N}\sum_{i=1}^N{\left( x_i-\mu _{MLE} \right) ^2} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{\left( x_{i}^{2}+\mu _{MLE}^{2}-2x_i\mu _{MLE} \right)} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}+\frac{1}{N}\sum_{i=1}^N{\mu _{MLE}^{2}-2\frac{1}{N}\sum_{i=1}^N{x_i\mu _{MLE}}}} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}+\mu _{MLE}^{2}-2\mu _{MLE}^{2}} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}-\mu _{MLE}^{2}} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}-\mu ^2-\left( \mu _{MLE}^{2}-\mu ^2 \right)} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}-\mu ^2} \right] -E\left[ \mu _{MLE}^{2}-\mu ^2 \right] $$
$$ E\left[ \sigma _{MLE}^{2} \right] =E\left[ \frac{1}{N}\sum_{i=1}^N{\left( x_i-\mu _{MLE} \right) ^2} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{\left( x_{i}^{2}+\mu _{MLE}^{2}-2x_i\mu _{MLE} \right)} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}+\frac{1}{N}\sum_{i=1}^N{\mu _{MLE}^{2}-2\frac{1}{N}\sum_{i=1}^N{x_i\mu _{MLE}}}} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}+\mu _{MLE}^{2}-2\mu _{MLE}^{2}} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}-\mu _{MLE}^{2}} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}-\mu ^2-\left( \mu _{MLE}^{2}-\mu ^2 \right)} \right] $$ $$ =E\left[ \frac{1}{N}\sum_{i=1}^N{x_{i}^{2}-\mu ^2} \right] -E\left[ \mu _{MLE}^{2}-\mu ^2 \right] $$
标签：inf nbsp src sum png ima 技术 图片 sig
原文地址：https://www.cnblogs.com/wangjs-jacky/p/10554884.html
