Problem 1
Question 11 of Chapter 3 in Introduction to Probability, Statistics, and Random Processes
The number of emails that I get in a weekday (Monday through Friday) can be modeled by a
Poisson distribution with an average of emails per minute.
The number of emails that I receive on weekends (Saturday and Sunday) can be modeled by a
Poisson distribution with an average of emails per minute.
a) What is the probability that I get no emails in an interval of length 4 hours on a Sunday?
b) A random day is chosen (all days of the week are equally likely to be selected), and a random
interval of length one hour is selected on the chosen day.
It is observed that I did not receive any emails in that interval.
What is the probability that the chosen day is a weekday?
a) Let be the number of emails received on a weekend, in a time interval of length minute.
Then has a Poisson distribution with parameter .
In Poisson’s Assumptions, the linearity assumption states
that the probability of an event occurring is proportional to the length of the time period.
As a consequence, the value of is proportional to the length of the time period.
And since the problem asked for a time period of length hours, we have that the
is now,
This should be intuitive because is the average number of occurences of an event in a time period .
Thus, if in minute, there is email, then in minutes (4 hours), there should be emails.
We can rephrase our initial statement as follows:
Let be the number of emails received on a weekend, in a time interval of length hours.
Then has a Poisson distribution with parameter .
Subsequently, the probability of getting no emails in a time interval of length hours is given by
b) Let be the number of emails received on a weekday, in a time interval of length hour, and
.
Let be the number of emails received on a weekend, in a time interval of length hour, and
.
Let be the disjoint union of and , and . is also a random variable
with a Poisson distribution . This is a consequence of the the additivity property
in Poisson’s Property 17.
Let be a random chosen day in a week, and is a random variable with a Uniform distribution
.
Now we can further decompose to , where is the random variable that
indicates whether the chosen day is a weekday, and is the random variable that indicates whether
the chosen day is a weekend. Note in particular that, is a random variable, a function
, where is the sample space, and is the range of , as indicated
in Definition 14.
The sample space is just the set of days in a week,
where we denote as .
Similarly, has a sample space that can be understood as the disjoint union of the sample spaces
of and . This is important as we want to invoke the Law of Total Probability.
Now we can formulate the problem as follows:
which means what is the probability of a chosen day is a weekday given that I did not receive any emails
in the time interval of hour of a random chosen week.
By Bayes’ Theorem (Definition 13), we have that
(13)
where ’s is derived from the Poisson formula when , because
since conditional probability shrinked
the sample space of to ; the is just the probability of a chosen day is a weekday.
The denominator is the probability of not receiving any emails in a time interval of length hour
of a random chosen week. This is not straightforward and we have to use the Law of Total Probability (Theorem 3).
That is,
With this, we have solved the problem,