STAT 100B HW 8 Due FridayProblem 1:SupposeY∼Bernoulli(λ). Suppose [X|Y= 0]∼N(μ0, σ2), and [X|Y= 1]∼N(μ1, σ2).(1) Find the marginal density ofX. Explain your result intuitively.(2) Calculate Pr(Y= 1|X=x). Explain your result intuitively.(3) Show that log[Pr(Y= 1|X=x)/Pr(Y= 0|X=x)] =β0+β1x, whereβ0andβ1can becalculated fromλ,μ1,μ0,σ2.Problem 2:SupposeYi∼Bernoulli(λ) independently, fori= 1, ..., n. Suppose [Xi|Yi= 0]∼N(μ0, σ2), and [Xi|Yi= 1]∼N(μ1, σ2). Find the maximum likelihood estimates ofλ,μ1,μ0,σ2.Problem 3:In the following data set, the first column records
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Maximum likelihood, Yi, Maximum likelihood estimates