MLE of a Normal Mean

Write the log-likelihood as a function of $\mu$ with $\sigma$ fixed.

Differentiate with respect to $\mu$ and set the result to zero.

1. 1. Write the log-likelihood

   The log-likelihood is $-\frac{1}{2{\sigma }^2}{\sum }_i(X_i-\mu {)}^2$ up to a constant not depending on $\mu$.

2. 2. Maximize

   Differentiating and setting to zero gives ${\sum }_i(X_i-\mu )=0$, so the maximizer is the sample mean.

   $$\hat{\mu }=\bar{X}=\frac{1}{n}\sum _i{X}_i$$

Answer: sample mean