Poisson Distribution Gauss file(s) basic_poisson.g Matlab file(s) basic_poisson.m A sample of T = 4 observations, yt = {6, 2, 3, 1}, is drawn from the Poisson distribution f(y; θ) = θ y exp[−θ] y! .

(a) Write the log-likelihood function, ln LT (θ).

(b) Derive and interpret the maximum likelihood estimator, θb.

(c) Compute the maximum likelihood estimate, θb.

(d) Compute the log-likelihood function at θb for each observation.

(e) Compute the value of the log-likelihood function at θb.

(f) Compute gt(θb) = d ln lt(θ) dθ θ=θb and ht(θb) = d 2 ln lt(θ) dθ2 θ=θb , for each observation. (g) Compute GT (θb) = 1 T X T t=1 gt(θb) and HT (θb) = 1 T X T t=1 ht(θb).

on, ln LT (θ).

(b) Derive and interpret the maximum likelihood estimator, θb.

(c) Compute the maximum likelihood estimate, θb.

(d) Compute the log-likelihood function at θb for each observation.

(e) Compute the value of the log-likelihood function at θb.

(f) Compute gt(θb) = d ln lt(θ) dθ θ=θb and ht(θb) = d 2 ln lt(θ) dθ2 θ=θb , for each observation. (g) Compute GT (θb) = 1 T X T t=1 gt(θb) and HT (θb) = 1 T X T t=1 ht(θb).