Let Y1, . . . , Yn iid∼ f (y; θ). For the following densities f , find:• the moment estimator of θ;• the MLE of θ;(a) Poisson: f (y; θ) = θy exp{−θ}y! for y ∈ {0, 1, 2, . . . } and θ > 0

1. Let Y1, . . . , Yniid∼ f (y; θ). For the following densities f , find:• the Fisher information I(θ) = −nE( d2d2θ log f (Yi; θ));• the Cramer-Rao lower bound for an unbiased estimator of θ.(a) Poisson: f (y; θ) = θy exp{−θ}y! for y ∈ {0, 1, 2, . . . } and θ > 0

the moment estimator of θ;• the MLE of θ

(d) Exponential: f (y; θ) = 1θ exp{− yθ } for y > 0 and θ > 0.

(a)The Poisson distribution truncated at zero has probability function givenbyf (x; θ) = exp(−θ)θxA(θ)x! , x = 1, 2, ... (θ > 0),whereA(θ) = 1 − exp(−θ).Let X1, . . . , Xn be a random sample of size n from this truncated Poissondistribution.(i) Show that this distribution belongs to the regular exponential distri-bution and give its canonical parameter c(θ) and the correspondingsufficient statistic T (X1, ..., Xn).

Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T . (i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameter regular exponential family, stating what the canonical parameter is equal to in terms of θ. (ii) Use the result in (i) to derive the maximum likelihood (ML) estimate of θ, θˆ = (μˆ, σˆ2)T , without the need to differentiate the log likelihood function with respect to θ.