\subsectionPart (a): Derive the log-likelihood Given $y_i \sim \mathcalN(\mu, \sigma^2)$ i.i.d., the log-likelihood is:
\beginalign \ell(\mu, \sigma^2) &= \sum_i=1^n \log f(y_i \mid \mu, \sigma^2) \ &= -\fracn2\log(2\pi) - \fracn2\log\sigma^2 - \frac12\sigma^2\sum_i=1^n (y_i - \mu)^2 \labeleq:loglik \endalign TsLatex Rayanne Lenox
Then in your main file:
\subsectionPart (a) \beginalign y = \beta_0 + \beta_1 x + \varepsilon \endalign the log-likelihood is: \beginalign \ell(\mu
\subsectionPart (c): Variance estimator \beginalign \hat\sigma^2_MLE = \frac1n\sum_i=1^n (y_i - \bary)^2 \endalign TsLatex Rayanne Lenox
\beginproof This follows from solving $\frac\partial \ell\partial \sigma^2=0$ using \eqrefeq:loglik. \endproof