During the pandemic a grocery delivery business started with the
policy of delivering orders with a maximum delay of 30 minutes (with respect
the desired delivery time picked by customers in their online purchases). Say
the delivery delay is preliminary modeled using a Uniform[0,30]-distributed
random variable.
a) What is the probability that a (random) order arrives during the last 5
minutes of the 30 minutes time window?
b) Is the probability in (a) the same that the probability that the order ar-
rives in the first 5 minutes (of the 30 minutes time window)?
Say after some months of operation data suggested that a better model for
the delivery delay time turned out to be a Normal distribution with mean 18
and standard deviation 6 minutes.
c) What is the probability that the order arrives after the 25th minute of
delay?
d) What is the probability that the order arrives before the time picked by
the customer?
e) Quantitatively comment on how well the business is doing with respect to
its policy.