Mathematics Homework Help

Problem 4 The weekly demand for a product has a normal distribution with mean 1, 000
and standard deviation 200. The current on hand inventory is 2, 200, and no deliveries
will be occurring in the next two weeks. Assume that the demands in different weeks are
independent.
(a) What is the probability that the demands in each of the next two weeks is less than
1, 100?
(b) What is the probability that the total of the demands in the next two weeks exceeds
2, 200?

Problem 5 Four runners, A, B, C and D are organized in two teams, with runners A and B
forming Team 1, and runners C and D forming Team 2. All four runners will run the same
distance. Suppose that the running times of the 4 runners are all exponentially distributed,
with respective means (not parameters!) µA = 10, µB = 12, µC = 8 and µD = 14 minutes.
Assume also that all four running times are independent.
(a) What is the distribution of the actual fastest time in each team?
(b) What is the probability that the winner of the race (the fastest one of the four runners)
will come from Team 2?

Problem 6 A system consists of three independent components, A, B and C, connected in
a series. The lifespans of components A, B, and C have an exponential distribution with
parameters 0.2, 0.1, and 0.15 (days)−1
(equivalently, means 5, 10, and 1/0.15 days). The
system fails when one of the components fails.
(a) Find the probability that the component whose failure caused the system to fail is
1
2
component C

(b) You can increase the expected lifespan of one of the components by 1 day. Which
component should you select for this improvement in order to cause maximal possible increase
in the expected lifespan of the entire system?