Answer question 1 below.

Let Y > 0 be a continuous random variable representing time from regimen start to bone-marrow transplant. Everyone does not survive long enough to get the transplant. Let X > 0 be a continuous random variable representing time from regimen start to death. We can assume X Y and model time to death as X Exp(rate = ) and time to transplant as Y Exp(rate = ). Where Exp(rate = ) denotes the exponential distribution with density f(z | ) = ez for z > 0 and 0 elsewhere - with > 0.

1.) We can define Z Exp(rate = ) as the measure of the heart function, which has an effect on time to death and transplant. We can fix our model to allow Z to affect the exponential rates: X | Z Exp(rate =1/Z ) and Y | Z Exp(rate = 1/Z). Assume there is conditional independence between X and Y given Z. (i.e. X Y | Z.) Find Cov(X, Y ). Hint: Your final answer should be in terms of , , and