Answer parts a and b of the question below. Show all work.

When diagnosing a disease (genetically) in a sample of patients. we let X {0, 1} be a random variable that is binary where X = 1 reflects disease presense, and X = 0 reflects absense of the disease. We are conducting a blood test that returns some continuous marker. A continuous variable expressing this marker is < Y < . We then use the following hierarchical model:

X Ber(p) Y | X N(₀(1 X) + ₁X , 2)

a. Find Cov(Y, X).

b. Find the marginal variance, V [Y], in terms of p, ₁, ₀, and ²