Derive MLEs and conditional Normal distributions

Suppose X1,…,Xn are i.i.d. Normal(μ, σ^2) and, independently, (X, Y) is one draw from a bivariate Normal with means μX, μY, variances σX^2, σY^2, and correlation ρ. Answer the following, showing all algebraic steps: (a) Write the pdf and cdf of Normal(μ, σ^2). Using them, express P(μ − σ ≤ X1 ≤ μ + σ) in terms of the standard Normal cdf Φ. (b) Derive the MLEs of μ and σ^2: (i) when σ^2 is known; (ii) when both μ and σ^2 are unknown; (iii) when μ is constrained so that μ ≥ 0 (give the closed-form MLE for μ and explain what happens when the unconstrained MLE is negative). (c) For the bivariate Normal (X, Y), derive the conditional distribution of X | Y = y: give its mean and variance and write the conditional pdf and cdf; then compute P(X > t | Y = y) in terms of Φ. (d) Is the usual MLE of σ^2 unbiased? If not, provide an unbiased estimator for σ^2 and relate it to the MLE.