Network logs track the origin of hits on a faculty web site. They show that hits
ID: 3220997 • Letter: N
Question
Network logs track the origin of hits on a faculty web site. They show that hits from inside UTD arrive according to a Poisson process with mean 30/day and hits from outside UTD arrive according to a Poisson process with mean 20/day, independent of the number of hits from inside UTD. Also, 50% of hits from inside UTD access problem solutions, but only 10% of hits from outside UTD access problem solutions. (a) What is the mean time between hits that access problem solutions? (b) What proportion of hits that access problem solutions come from inside UTD? (c) Given there were 50 hits over 2 days from inside UTD that accessed problem solutions, what is the expected total number of hits over 2 days? If a hit did not access problem solutions, what is the probability it came from outside UTD?Explanation / Answer
i = 30 Mean hits from inside UTD per day o = 20 Mean hits from outside UTD per day a) We find the number of hits from inside or outside UTD which access problem solutions = 30 x 0.5 + 20 x 0.1 = 17 Thus 17 hits access problem solutions in a day Considering a day with 24 hours, we get Time between two hits = 24/17 hours = 1.412 hours Hence Mean time between hits that access problem solutions = 1.412 hours b) From (a), we know 17 hits access problem solutions in a day Hits that access problems solutions from inside UTD = 30 x 0.5 = 15 Proportion = 15/17 = 0.8824 Proportion of hits that access problem solutions coming from inside UTD = 0.8824 c) There were 50 hits over 2 days from inside UTD that accessed problem solutions 50% of hits from inside UTD access problem solutions We have to find i which is the expected total number of hits over 2 days from inside UTD 0.5 x = 50 i = 100 Expected total number of hits over 2 days from outside UTD = 20 x 2 = 40 (Given per day hits outside UTD = 20) Total hits = 100 + 40 = 140 Expected total number of hits over 2 days = 140 To find P(hit came from outside UTD/hit did not access problem solution) = P(hit came from outside UTD hit did not access problem solution) / P(hit did not access problem solution) = P(hit came from outside UTD and hit did not access problem solution) / P(hit did not access problem solution) We know 30 hits come from inside UTD and 20 hits come from outside UTD Thus P(hit came from outside UTD) = 20 / (30+20) = 20/50 = 0.4 We know P(hits outside UTD access problem solutions) = 0.10 Hence P(hits outside UTD and did not access problem solutions) = 1 - 0.10 = 0.9 Thus P(hit came from outside UTD and hit did not access problem solution) = 0.4 * 0.9 = 0.36 ………. (A) Also P(hit did not access problem solution) = P(hit inside UTD did not access problem solution ) and P(hit outside UTD did not access problem solution) = (1 - 0.5) x (1-0.1) = 0.5 x 0.9 = 0.45 ………(B) P(hit came from outside UTD and hit did not access problem solution) / P(hit did not access problem solution) = (A) / (B) = 0.36/0.45 = 0.8 P(hit came from outside UTD/hit did not access problem solution) = 0.8