1. Attempt all problems. Show all work. Upload to gradescope by 3:30pm.

2. The problem weights are as given in the parentheses.

3. You are allowed one 8.5 by 11 sheet of paper with notes of your choice on both sides.

4. If you use Matlab or R programs state explicitly how you called them.

5. No communication about the exam is allowed with anybody except the proctor.

P1.  (30) Suppose Bob’s email inbox is empty at time 0.  An incoming email

message can be real or spam.  Bob gets 8 spam messages per hour, and 5 real messages per hour.  Suppose the arrival processes of the spam and real emails are independent Poisson processes.  Currently Bob does not have any spam filter, so both type of emails are put in the inbox.

(a)  (4) What is the distribution of the time when the first message arrives (it could be either real or spam) in the inbox?

(b)  (4) What is the probability that first message is spam?

(c)  (5) What is expected time until the arrival of the third real message?

(d)  (5) What is the variance of the total number of messages over four hours?

Bob gets annoyed by the spam messages and decides to add a spam filter. The spam filters blocks 80% of the spam messages and puts them in the spam folder.  Unfortunately, it also blocks 10% of ral messages and puts them in the same spam folder. The spam folder is empty at time 0.

(e)  (6) Let N(t) be the number of messages in the spam folder at time t. Is {N(t),t ≥ 0} a Poisson process? Why or why not? If it, what is its rate?

(f)  (6) If 20 spam messages arrive in the spam folder in the first four

hours, what is the probability that 3 of them are real messages?

P2.  (30) Mrs.  Smith suffers from a chronic pain condition that requires fre-

quent visits to the doctor.  She has an option of visiting the doctor in person or remotely. The in-person visits are more effective, but more in- convenient; while remote visits are convenient but not as effective. After a remote visit Mrs. Smith needs to revisit the doctor after about 25 days, while after in-person visit she needs to return in about 40 days. Suppose she opts for in-person visit about 30% of the visits, regardless of past his- tory.  Let X(t) be 1 if Mrs.  Smith’s most recent visit up to time t was in-person, and 2 if the most recent visit up to time t was remote. Assume the initial visit was in-person at time 0.