123
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$$ Eleft[ sigma _{MLE}^{2} ight] =Eleft[ frac{1}{N}sum_{i=1}^N{left( x_i-mu _{MLE} ight) ^2} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{left( x_{i}^{2}+mu _{MLE}^{2}-2x_imu _{MLE} ight)} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}+frac{1}{N}sum_{i=1}^N{mu _{MLE}^{2}-2frac{1}{N}sum_{i=1}^N{x_imu _{MLE}}}} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}+mu _{MLE}^{2}-2mu _{MLE}^{2}} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}-mu _{MLE}^{2}} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}-mu ^2-left( mu _{MLE}^{2}-mu ^2 ight)} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}-mu ^2} ight] -Eleft[ mu _{MLE}^{2}-mu ^2 ight] $$
$$ Eleft[ sigma _{MLE}^{2} ight] =Eleft[ frac{1}{N}sum_{i=1}^N{left( x_i-mu _{MLE} ight) ^2} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{left( x_{i}^{2}+mu _{MLE}^{2}-2x_imu _{MLE} ight)} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}+frac{1}{N}sum_{i=1}^N{mu _{MLE}^{2}-2frac{1}{N}sum_{i=1}^N{x_imu _{MLE}}}} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}+mu _{MLE}^{2}-2mu _{MLE}^{2}} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}-mu _{MLE}^{2}} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}-mu ^2-left( mu _{MLE}^{2}-mu ^2 ight)} ight] $$ $$ =Eleft[ frac{1}{N}sum_{i=1}^N{x_{i}^{2}-mu ^2} ight] -Eleft[ mu _{MLE}^{2}-mu ^2 ight] $$
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