Question

a) Given the joint pdf of the continuous RVs X and Y: fxy(x, y) = c for the region {0 sxs y,0 s y 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 given Y, the support of the random varaible X is [0,Y]. Thus, with Y the support of the condiional distribution of X given Y changes and hence the conditional distribution itself changes. This suggests that X and Y are not independent.

b)

Let S be the event that a randomly chosen person is smoker and ~S be the event that she/he is not. Then, we are given that

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

Thus, we get the probability that a randomly selected person has illness

= P(A)

=P(A|S)P(S)+P(A|~S)P(~S)

= 0.1x0.2+0.005x0.8

= 0.024 (Ans.)