Mixture distribution suppose that the joint distribution of the two random variables x and y is

a. Find the maximum likelihood estimators of β and θ and their asymptotic joint distribution.

b. Find the maximum likelihood estimator of θ/(β + θ) and its asymptotic distribution.

c. Prove that f (x) is of the form f (x) = γ (1 ?? γ)x, x = 0, 1, 2, . . . , and find the maximum likelihood estimator of γ and its asymptotic distribution.

d. Prove that f (y | x) is of the form Prove that f (y | x) integrates to 1. Find the maximum likelihood estimator of λ and its asymptotic distribution. [Hint: In the conditional distribution, just carry the xs along as constants.]

e. Prove that f (y) = θe??θy, y ?? 0, θ>0. Find the maximum likelihood estimator of θ and its asymptotic variance.

f. Prove that Based on this distribution, what is the maximum likelihood estimator of β?

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f(x, y) = @e-(8+)y (By)* x! f(y|x) = f(x|y) = B,9>0, y ≥ 0, x = 0, 1, 2,.... Part (d) λe-^(^y)* x! Part (e) e-By (By).* x! y ≥ 0, λ > 0. x = 0, 1, 2, ..., ß> 0.