Question

There are two types of calls to the MIT Campus Patrol. Type A calls (distress calls) arrive as a Poisson process
with rate 5 calls per hour. Type B calls (professors who have lost their keys) arrive as an independent Poisson
process with rate 2 calls per hour. Let us fix t to be 12 o’clock.
a) What is the expected length of the interval that t belongs to? (That is, the interval from the last event
before t until the first event after t.)
b) What is the probability that t belongs to an AA interval? (That is, the first event before, as well as the
first event after time t are both of type A.)
c) What is the probability that between t and t + 60 minutes, we have exactly two events, one of type A,
followed by one of type B?

Answer 1

a) All together, 5+2 ie 7 calls arrive per hour. So, expected length of interval that t belongs to is 60/7 ie 8.5 minutes.

c) Both events have exactly one arrival each in an hour and have frequently each equal to one.

Its solution has been attached.

b) Conditions are the first event should be of A-type and just after that, the 2nd event should be of A-type too.