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. For an event with probability p, the odds of the event is defined as p/1p. With a higher odds, an event is more likely. With the odds of 1 indicates the event is as likely to occur as not likely. i.e. we are maximally uncertain about the status of the diseae. Find the conditional odds of disease given Y , P(X=1|y)/(1P (X=1|y)). What is the odds of disease if ₁ = ₀? What is the interpretation behind this result?