Question

Question 1) (60 Marks) a) Given the joint pdf of the continuous RVS X and Y: fxy(x, y) = c for the region {0 sxs y,o sy s 1} and zero elsewhere.Where “c” is a constant. Determine if the RV X and Y are independent. (30 Marks) b) Given the following information. For a certain form of illness, (call the illness “A”), the likelihood of having the illness is 0.1 if the person is a smoker and 0.005 if the person is not a smoker. In a doctor's office 20% of the patients are smokers. What is the probability that a randomly selected patient in the office has the illness? (30 Marks)

Answer 1

a) Note that the support of the conditional distribution of X given Y depends on Y, and thus the RVs are not independent.

b) Let S denote the event that a person is smoker and ~S denote its conplement.

Then, we have P(S) = 0.2, P(~S) = 0.8, P(A|S) = 0.1, P(A|~S) = 0.005

We are required to find

P(A) = P(A|S)P(S)+P(A|~S)P(~S) [Law of total probability]

= 0.1x 0.2 + 0.005x0.8 = 0.024 (Ans.)