Exercises
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Exercises#
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 \(\frac{1}{6}\) 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 \(\frac{1}{30}\) 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 \(X\) be the number of emails received on a weekend, in a time interval of length \(1\) minute. Then \(X \sim \text{Poisson}(\frac{1}{30})\) has a Poisson distribution with parameter \(\lambda = \frac{1}{30}\).
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 \(\lambda\) is proportional to the length of the time period.
And since the problem asked for a time period of length \(4\) hours, we have that the \(\lambda\) is now,
This should be intuitive because \(\lambda\) is the average number of occurences of an event in a time period \(T\). Thus, if in \(1\) minute, there is \(\frac{1}{30}\) email, then in \(240\) minutes (4 hours), there should be \(8\) emails.
We can rephrase our initial statement as follows:
Let \(X\) be the number of emails received on a weekend, in a time interval of length \(4\) hours. Then \(X \sim \text{Poisson}(8)\) has a Poisson distribution with parameter \(\lambda = 8\).
Subsequently, the probability of getting no emails in a time interval of length \(4\) hours is given by
b) Let \(X\) be the number of emails received on a weekday, in a time interval of length \(1\) hour, and \(X \sim \poisson \lpar 60 \times \frac{1}{6} \rpar = \poisson(10)\).
Let \(Y\) be the number of emails received on a weekend, in a time interval of length \(1\) hour, and \(Y \sim \poisson \lpar 60 \times \frac{1}{30} \rpar = \poisson(2)\).
Let \(Z\) be the disjoint union of \(X\) and \(Y\), and \(Z = X \sqcup Y\). \(Z\) is also a random variable with a Poisson distribution \(Z \sim \poisson(12)\). This is a consequence of the the additivity property in Poisson’s Property 17.
Let \(W\) be a random chosen day in a week, and \(W\) is a random variable with a Uniform distribution \(W \sim \uniform \lpar 1, 7 \rpar\).
Now we can further decompose \(W\) to \(W = W_1 \sqcup W_2\), where \(W_1\) is the random variable that indicates whether the chosen day is a weekday, and \(W_2\) is the random variable that indicates whether the chosen day is a weekend. Note in particular that, \(W\) is a random variable, a function \(W: \S \to \R\), where \(\S\) is the sample space, and \(\R\) is the range of \(W\), as indicated in Definition 14.
The sample space \(\S\) is just the set of days in a week, where we denote as \(\S = \lset 1, 2, 3, 4, 5, 6, 7 \rset\).
Similarly, \(Z\) has a sample space that can be understood as the disjoint union of the sample spaces of \(X\) and \(Y\). 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 \(1\) hour of a random chosen week.
By Bayes’ Theorem (Definition 13), we have that
where \(\defb\)’s \(e^{-10}\) is derived from the Poisson formula when \(\lambda = 10\), because \(\P \lsq Z = 0 \lvert W_1 \rsq = \P \lsq X = 0 \rsq = e^{-10}\) since conditional probability shrinked the sample space of \(Z\) to \(X\); the \(\frac{5}{7}\) is just the probability of a chosen day is a weekday.
The denominator \(\P \lsq Z = 0 \rsq\) is the probability of not receiving any emails in a time interval of length \(1\) 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,