Generate a random sample of n 1 000 for theta 3 compute

Question

1. For each of the following distributions, generate 1,000 variables and compare the mean, variance and histogram of the generated random variables with the theoretical values.

(a) Y ∼ binomial(8, 2/3)
(b) Y ∼ poisson(2/3)
(c) Y ∼ gamma(2, 2)
(d) Y ∼ beta(2, 4)

2. Let {X_n }^N _n=1 be a random sample from a continuous distribution with pdf:

fx_n (x_n ;θ) = θx^θ-1 _n I{x_n , ∈ (0, 1) }, θ ∈ Θ ≡ R₊₊

(a) Find the method of moments estimator and the maximum likelihood estimator.

(b) Generate a random sample of N = 1, 000 for θ = 3, compute the method of moments and the maximum likelihood estimators. Compare them with the true population parameter.