Derive MLEs and conditional Normal distributions

Assume $X_1, \ldots, X_n$ are i.i.d. $\mathrm{Normal}(\mu, \sigma^2)$, and that $(X, Y)$ is an independent single draw from a bivariate Normal distribution with means $\mu_X, \mu_Y$, variances $\sigma_X^2, \sigma_Y^2$, and correlation $\rho$. Answer all parts below, showing the algebra at each step:

(a) Give the probability density function and cumulative distribution function of $\mathrm{Normal}(\mu, \sigma^2)$. Use them to express $P(\mu - \sigma \le X_1 \le \mu + \sigma)$ in terms of the standard Normal cdf $\Phi$.

(b) Derive the MLEs of $\mu$ and $\sigma^2$: (i) when $\sigma^2$ is known; (ii) when both $\mu$ and $\sigma^2$ are unknown; (iii) when $\mu$ is constrained by $\mu \ge 0$ (give the closed-form MLE for $\mu$ and explain what happens when the unconstrained MLE is negative).

(c) For the bivariate Normal pair $(X, Y)$, derive the conditional distribution of $X \mid Y = y$: give its mean and variance and write the conditional pdf and cdf; then compute $P(X > t \mid Y = y)$ in terms of $\Phi$.

(d) Is the usual MLE of $\sigma^2$ unbiased? If not, provide an unbiased estimator for $\sigma^2$ and relate it to the MLE.

Overview: This problem assesses skill in parametric inference and probability, including maximum likelihood estimation, estimator bias, and conditioning in univariate and bivariate Normal distributions, as well as constrained parameter estimation.