MATH 351 Homework 06

Suppose you have independent random variables $X_1, \ldots, X_N$ from the Normal distribution. The Normal distribution has density function $$f(x | \mu, \sigma) = (2\pi\sigma^2)^{-1/2} \exp{(-(x - \mu)^2 / 2\sigma^2)}$$ where the parameter vector to estimate has two values, $\theta = (\mu, \sigma)$. Find the maximum likelihood estimators for both $\mu$ and $\sigma$. Type up your solutions using $\LaTeX$ in a Google Colab notebook.