An electronic device is designed to switch house lights on and off at random times after it has been activated. Suppose that the device is designed in such a way that it will be switched on and off exactly once in a 1-hour period. Allow Y denote the time at which the lights are turned on and X the time they are turned off. Suppose that the joint density for (X,Y) is given by fXY(x,y) - 8xy for 0

(A) Verify that fXY satisfies the situations necessary to be a density.

(B) Find out E[XY].

(C) Find out the probability that the lights will be switched on in 1/2 hour after being activated and then switched off again in 15 mins.

(D) Find out the marginal density for X. Find E[X] and E[X2].

(E) Find out the marginal density for Y. Find E[Y] and E[Y2].

(F) Are X and Y independant?

(G) Find out the conditional distribution of X given Y.

(H) Find out the probability that the lights will be switched off within 45 mins of the system being activated given that they were switched on 10 mins after the system was activated.

(I) Find out the curve of regression of X on Y. Is the regression linear?

(J) Find out the expected time that the lights will be turned off given that they were turned on 10 mins after the system was activated.

(K) Based upon the physical description of the problem, would you expect ρ to be positive, negative, or 0? Explain and verify by computing ρ.