Normal Distribution Consider a random sample of size T, {y1, y2, · · · , yT }, of iid random variables from the normal distribution with unknown mean θ and known variance σ 2 0 = 1 f(y; θ) = 1 √ 2π exp  − (y − θ) 2 2  .

(a) Derive expressions for the gradient, Hessian and information matrix.

(b) Derive the Cram´er-Rao lower bound.

(c) Find the maximum likelihood estimator θb and show that it is unbiased. [Hint: what is R ∞ −∞ yf(y)dy?]

(d) Derive the asymptotic distribution of θb.

(e) Prove that for the normal density E  d ln lt dθ  = 0 , E d ln lt dθ 2  = −E  d 2 ln lt dθ2  .

(f) Repeat parts

(a) to

(e) where the random variables are from the exponential distribution f(y; θ) = θ exp[−θy] .