Answer part a 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. Suppose p = .5. Find an expression for P(X = 1 | y) only in terms of f_Y|X(y | X = 1) and

f_Y|X(y | X = 0), where f_{Y |X} represents the conditional normal density given above evaluated at some X = x.