Jointly Distributed random variables?

annie and alvie have agreed to meet between 5pm and 6pm for dinner at a health food restaurant. x=annies arival time, y=alvies arrival time. x and y are independent with each uniformly distributed on the interval [5,6].





a) whats the joint pdf of x and y?





b)whats the probability they both arrive between 5:15 and 5:45?





c) if the first one to arrive with only wait 10 min before leaving, whats the probability that they have dinner at the health food restaurant? [hint: the event of interest is A={(x,y): |x-y|%26lt;=1/6}.]

Jointly Distributed random variables?
a) the pdf for a uniform distribution over an interval [a,b] is





f(x) = 1/(b-a) = 1 in your case since 6-5=1.





Siince they are both independent then the joint pdf is the product of the 2 pdf s . Each pdf is 1 over the whole inerval so the joint pdf g(x) is also 1*1 = 1 for all x in [5,6] and 0 otherwise.





b) looks good.





c) the area in question is bounded by the area: x-1/2 %26lt; y and y%26lt; x + 1/6. This will give you teh complement of what you calculated. Ans I have is .30555 (that's 1 - your answer) which makes more sense than the bigger probability you have. To make things simple, use the interval [0,1] to draw your lines instead of [5,6]. The correct area is the small diagonal rectangle plus the two little triangles next to it (coordinates (0,1/6), (1/6,0), (0,0), and (1,1), (5/6,1) , (1,5/6)