Question

Annie and Alvie have agreed to meet between 5:00 P.M. and 6:00 P.M. for dinner at a local health-food restaurant. Let X = Annie's arrival time and Y = Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6]. (a) What is the joint pdf of X and Y? f(x,y) = Correct: Your answer is correct. 5 ≤ x ≤ 6, 5 ≤ y ≤ 6 Correct: Your answer is correct. otherwise (b) What is the probability that they both arrive between 5:30 and 5:45? (c) If the first one to arrive will wait only 20 min before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A = (x, y): |x − y| ≤ 1/3 . ] (Round your answer to three decimal places.)

Answer 1

Annie and Alvie have agreed to meet between 5:00 P.M. and 6:00 P.M. for dinner at a local health-food restaurant. Let X = Annie's arrival time and Y = Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].

(a) The joint pdf of X and Y is

Since X and Y are independent, we multiply the individual pdf's together. f(x,y)= f(x)f(y).

(b) The probability that they both arrive between 5:30 and 5:45 is define as

divide the minutes into the decimal by using (1/60) times that means 5:30 means 5.(30/60) = 5.5

(c) We can use the condition,

So, the area under this range is

It consists of the whole square minus two right triangles of side 2/3 on the upper left and bottom right. The remaining area is therefore equal to