Final answer:
In the given scenario with two independent customers arriving at uniformly random times between 2 pm and 8 pm, the expected time of the first arrival is 3 pm, and the expected time of the last arrival is 5 pm.
Step-by-step explanation:
The question subject is probability and statistics, a topic within mathematics. The query concerns determining expected arrival times under a uniform distribution. Two customers are set to arrive at random between 2 pm and 8 pm. To calculate the expected times for the first and last arrivals is an exercise in understanding continuous uniform distribution.
Expected Time of First Arrival
For the first (earlier) arrival, we denote the arrival times of the two customers as X and Y and suppose they are uniformly distributed between 2 pm (t=0) and 8 pm (t=6). The minimum of two uniformly distributed variables, min(X, Y), is expected to arrive earlier than the midpoint of t=0 and t=6. By integrating the probability density functions of the two independent variables over their range and finding the expected value of the minimum, we can find the expected first arrival time is at 3 pm.
Expected Time of Last Arrival
For the last (later) arrival, using the maximum of two uniformly distributed variables, max(X, Y), we similarly calculate the expected value of the maximum. The expected last arrival time is at 5 pm.