Question

1. Consider the differential equation dy/dt = 0.2y(t) -2, y(0) = 40. (a) Showing your work, calculate y(0.2) using Euler's Method and a step size of At = 0.1 (b) Repeat with At = 0.2 and compare your answer with (a) 2. Consider the number of phone calls in a minute to a corporate Delta Airlines (800) number is Poisson distributed with a mean of 1. = 7.5 during the busy hour. (a) What is the probability of no calls in a minute? (b) What is the probability of more than 3 calls in a minute? 3. A financial services firm deploys two separate gateway routers, named Zebra and Unicorn, to connect to their ISP in a redundant fashion. If the Zebra gateway fails, traffic can be processed through the Unicorn gateway, and vice versa. During the past year, data has been collected on the number of crashes and time between failures of Zebra and Unicorn gateway for a reliability analysis as shown in the table below. Zebra Unicorn Mean Standard Deviation Mean Standard Deviation Time between failures (min) 185.23 187.2 205.15 160.3 Number of crashes 1024 840 a. Determine a 95% confidence interval on the mean time between failures of the Zebra gateway b. Determine a 95% confidence interval on the mean time between failures of the Unicorn gateway C. Does the data support the conclusion that Unicorn is more reliable than Zebra?

Answer 1

2. No of phone calls in 60 mins = 7.5

So no of phone calls in 1 min = 7.5/60 = 0.125

Let X= no of calls in a min

So, X~Poisson(0.125) distribution

PDF of X is given as: P(X=x) = e^-0.125*(0.125)^x/x!

a) We want, P(X=0) = e^-0.125 = 0.8825

b) P(X>3) = 1-P(X<=3) = 1- P(X=0) - P(X=1) - P(X=2) - P(X=3)

Putting these values of x in the PDF above and calculating the respective probabilies we get,

P(X>3) = 0.00001

3. a) Since n= 1024 is a very large sample we can use z score for the ci calculation. 95% CI for the mean u is given as:

Sample mean +-z*sample SD/√n

= 185.23 +-1.96*187.2/√1024

= (173.764, 196.696)

b) similarly 95% CI for Unicorn u is given as:

205.15 +- 1.96*160.3/√840

= (194.309, 215.991)

c) Since from the CI we can see that the mean time between failures is higher for Unicorn gateway, it seems more reliable than Zebra at the 5% significance level.