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 . (a) What is the joint pdf of X and Y? f(x, y) = . 5 ≤ x ≤ 6, 5 ≤ y ≤ 6 (b) What is the probability that they both arrive between 5:15 and 5:45? (c) If the first one to arrive will wait only 5 min before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant?

(a) f(x,y)=1 for 5≤x≤6 and 5≤y≤6 and 0 otherwise,  (c) 23/144

Step-by-step explanation:

(a) Let X=Annie's arrival time and Y = Alvie's arrival time.

Since X and Y is independent with each other, we can simply write as

(b) Since X and Y independent from each other we can simply write

and from the distribution of X and Y (given as uniform between 5:00 PM and 6:00 PM), we can write as

Therefore,

0.25 is the probability that they both arrive between 5:15 and 5:45.

(c)  Simply the event of interest can be defined as

This situation can be visualized as given in the attachment:

The region of interest (probability of the event) is region II.

$32,660

step-by-step explanation: