Question

Suppose customers arrive at a 7−11 convenience store according to a Poisson process with a rate of 10 customers per hour. Compute the following probabilities:

1. No customers in an hour
2. Exactly 5 customers in an hour
3. Exactly 10 customers in 2 hours
4. At least two customers in half an hour
5. Compare the answers you found in (2) and (3) and explain why they are different.

Answer 1

The probabilities associated with the Poisson process and exponential distribution at a 7-11 convenience store include calculations for no arrivals, exact counts of arrivals in certain hours, and understanding the effect of differing time intervals. It involves using the characteristics of the Poisson and exponential distributions to determine waiting times and probabilities of certain numbers of events occurring.

Step-by-step explanation:

The customers arriving at a 7-11 convenience store according to a Poisson process with a rate of 10 customers per hour involves calculating various probabilities based on this distribution:

1. The probability of no customers in an hour.
2. The probability of exactly 5 customers in an hour.
3. The probability of exactly 10 customers in 2 hours.
4. The probability of at least two customers in half an hour.

Additionally, the student asked to compare the probabilities found in parts 2 and 3 and explain why they are different. This involves understanding the Poisson process and the effect of different time intervals on the probability of a certain number of arrivals.

For computational examples relevant to the Poisson process:

• The probability of more or fewer arrivals given specific time frames such as an hour, two hours, or half an hour.
• The differences in probabilities when comparing different time intervals.

For the exponential distribution:

• Understanding the exponential distribution, which models the time spent waiting between events in a Poisson process.
• Calculating probabilities related to the time between successive events or arrivals.

Lastly:

1. Which type of distribution the Poisson model can approximate and when this approximation is appropriate.