Question

Alice and Bob meet for lunch every day at a random time between noon and 1 p.m. The joint p.d.f. of (x), the time Alice arrives (in minutes after 12 p.m.), and (y), the time Bob arrives (in minutes after 12 p.m.), is given by (f(x, y)). If Bob always arrives after Alice, what is the relationship between the arrival ×?

a) (x < y)

b) (x = y)

c) (x > y)

Answer 1

The relationship between Alice's and Bob's arrival times is (x < y), indicating Alice arrives before Bob. Probabilities of arrival within certain time frames can be determined by integrating the joint p.d.f over the specified intervals.

Step-by-step explanation:

If Alice and Bob meet for lunch and Bob always arrives after Alice, then the relationship between their arrival times is (x < y), where x represents the time Alice arrives and y represents the time Bob arrives, both minutes after 12 p.m. This means that Alice's arrival time is always less than Bob's arrival time.

To find the probability of certain time intervals, we would integrate the joint probability density function (p.d.f.) over the region of interest. For instance, to find the probability that Alice arrives at most 30 minutes after 12 p.m., we would integrate the p.d.f. from 0 to 30 on the x-axis and from x to 60 on the y-axis, since Bob can arrive anytime after Alice up until 1 p.m. The area under the curve in the shade would represent the desired probability. Similarly, the probability that Alice arrives between 30 and 40 minutes after 12 p.m. would be found by integrating the p.d.f. over the respective range for Alice and the corresponding range for Bob.