This question is based on a time when people needed to log in to email servers to write emails. Assume that Adam logs in to an email server at time zero. He then spends his time exclusively on writing emails. The times that his emails are sent can be modeled by a Poisson process with a rate
λ A
/
hour. His friend Brianna also logs in to the server at time 1 (which means 1 hour later) and starts typing emails independently of Adam. The time that her emails are sent also can be modeled by a Poisson process with a rate
λ B
/
hour. (a) (6 points) Let
Y 1
, and
Y 2
be the times at which Adam's first and second emails were sent. Find
E[Y 2
∣Y 1
]
(b) (8 points) Similar to the last question, let
Y 1
and
Y 2
be the times at which Adam's first and second emails were sent. When Brianna logs in to the server, you are told that Adam has sent exactly one email so far. (b.1) (4 points) What is the conditional expectation of
Y 1
given this information? (b.2) (4 points) What is the conditional expectation of
Y 2
given this information? (c) (8 points) Find out the PMF of the total number of emails sent by two of them together during the interval of
[0,2]
(which means in the first two hours). (d) (8 points) Given that a total of 10 emails were sent during the period of
[0,2]
(the first two hours), what is the probability that exactly four emails were sent by Adam? (e) (10 points) Suppose that
λ A
=4
. Use Chebyshev's (4 points) and Markov's (4 points) inequality to find upper bounds on the probability that Adam sent at least five emails during the time interval [0, 1 ]. Which one provides a better bound ( 2 points)?
