Let X₁,X₂, . . .,X_n be a random sample from each of the following distributions involving the parameter θ. In each case find the mle of θ and show that it is a sufficient statistic for θ and hence a minimal sufficient statistic.
(a) b(1, θ), where 0 ≤ θ ≤ 1.
(b) Poisson with mean θ > 0.
(c) Gamma with α = 3 and β = θ > 0.
(d) N(θ, 1), where −∞ < θ < ∞.
(e) N(0, θ), where 0 < θ < ∞.