(Solution) - Mixture distribution suppose that the joint distribution -(2025 Original AI-Free Solution)

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Mixture distribution suppose that the joint distribution of the two random variables x and y is
a. Find the maximum likelihood estimators of ? and ? and their asymptotic joint distribution.
b. Find the maximum likelihood estimator of ?/(? + ?) and its asymptotic distribution.
c. Prove that f (x) is of the form f (x) = ? (1 ? ?)x, x = 0, 1, 2, . . . , and find the maximum likelihood estimator of ? 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 ? and its asymptotic distribution. [Hint: In the conditional distribution, just carry the xs along as constants.]
e. Prove that f (y) = ?e??y, y ? 0, ?>0. Find the maximum likelihood estimator of ? and its asymptotic variance.
f. Prove that Based on this distribution, what is the maximum likelihood estimator of ??