1. Determine the maximum likelihood estimator for μ from an exponential random sample X₁, . . . ,X_N with common mean EX₁ = μ. Make sure to verify that your estimator indeed maximises the likelihood.

2. Determine an exact 1 -α stochastic condence interval for λ based on an exponential random sample X₁, . . . ,X_N ~ ∈iid Exp(λ).

3. Give an appropriate test-statistic T, its distribution under H₀, and a critical region C for the two-sided hypothesis test H₀, λ = λ₀ vs H₁. λ = λ₀, assuming under H₀ that data is distributed as X₁, . . . ,X_N ~ ∈iid Exp(λ₀).

4. The least-squares solution β^ for a linear regression model is β^= (β^ ₀, β^₁ )^T , where

Verify that β^ solves the normal equations A^TAβ  = A^Ty for data y.

5. For an outcome Y = y of a linear model Y = Aβ + ∈ ", where " ∈ ~ N(0, σ²I), show that the least-squares solutionβ^solving the normal equations A^TAβ  = A^Ty is the maximum likelihood estimator for β.

6. For a linear regression model.

(a) Determine the joint distribution of the least-squares solution β^.

(b) Using (a), determine the distribution of the linear regression line at an arbitrary explanatory point x, that is, the distribution of Y^ where Yβ^ = β₀^ + β^₁x.

Comment on how to go about constructing a condence interval for the linear regression line at an arbitrary explanatory point x.