Normal likelihood
= Normal likelihood
{title2=$L(\mu,\sigma^2)=(2\pi\sigma^2)^{-n/2}\exp[-\sum_i(x_i-\mu)^2/(2\sigma^2)]$}
The <likelihood function> of independent observations from the same <normal distribution>. For fixed positive <variance> $\sigma^2$, its unique maximizing mean is the <sample mean>: $\sum_i(x_i-\mu)^2=\sum_i(x_i-\overline x)^2+n(\mu-\overline x)^2$. This algebraic identity proves global optimality without relying only on a stationary derivative.