A Markov chain model for customer visits to an auto repair...

A Markov chain model for customer visits to an auto repair shop is described in the article "Customer Lifetime Value Prediction by a Markov Chain Based Data Mining Model: Application to an Auto Repair and Maintenance Company in Taiwan" (Scientia Iranica, 2012 : $849-855$ ). Customers make between 0 and 4 visits to the repair shop each year; for any customer that made exactly $i$ visits last year, the number of visits s/he will make next year follows a Poisson distribution with parameter $\mu_{i}$ . (The event "4 visits" is really "4 or more visits," so the probability of 4 visits next year is calculated as $1-\sum_{x=0}^{3} p\left(x ; \mu_{i}\right)$ from the appropriate Poisson pmf.) Parameter values cited in the article, which were estimated from real data, appear in the accompanying table. $$ \begin{array}{c|cccc}{i} & {0} & {1} & {2} & {3} & {4} \\ \hline \mu_{i} & {1.938696} & {1.513721} & {1.909567} & {2.437809} & {3.445738}\end{array} $$ $$ \begin{array}{l}{\text { (a) Construct the one-step transition matrix for the chain } X_{n}=\text { number of repair shop visits by }} \\ {\text { a randomly selected customer in the } n \text { th observed year. }} \\ {\text { (b) If a customer made two visits last year, what is the probability that s/he makes two visits }} \\ {\text { next year and two visits the year after that? }}\end{array} $$ $$ \begin{array}{l}{\text { (c) If a customer made no visits last year, what is the probability s/he makes a total of exactly }} \\ {\text { two visits in the next } 2 \text { years? }}\end{array} $$

A Markov chain model for customer visits to an auto repair shop is described in the article
"Customer Lifetime Value Prediction by a Markov Chain Based Data Mining Model: Application to an Auto Repair and Maintenance Company in Taiwan" (Scientia Iranica, 2012 :
$849-855$ ). Customers make between 0 and 4 visits to the repair shop each year; for any customer that made exactly $i$ visits last year, the number of visits s/he will make next year follows a
Poisson distribution with parameter $\mu_{i}$ . (The event "4 visits" is really "4 or more visits," so the
probability of 4 visits next year is calculated as $1-\sum_{x=0}^{3} p\left(x ; \mu_{i}\right)$ from the appropriate Poisson pmf.) Parameter values cited in the article, which were estimated from real data, appear in the accompanying table.
$$
\begin{array}{c|cccc}{i} & {0} & {1} & {2} & {3} & {4} \\ \hline \mu_{i} & {1.938696} & {1.513721} & {1.909567} & {2.437809} & {3.445738}\end{array}
$$
$$
\begin{array}{l}{\text { (a) Construct the one-step transition matrix for the chain } X_{n}=\text { number of repair shop visits by }} \\ {\text { a randomly selected customer in the } n \text { th observed year. }} \\ {\text { (b) If a customer made two visits last year, what is the probability that s/he makes two visits }} \\ {\text { next year and two visits the year after that? }}\end{array}
$$
$$
\begin{array}{l}{\text { (c) If a customer made no visits last year, what is the probability s/he makes a total of exactly }} \\ {\text { two visits in the next } 2 \text { years? }}\end{array}
$$

Key Concepts

Multi-step Transition Probabilities

Multi-step transition probabilities in a Markov chain refer to the probabilities of transitioning from one state to another over multiple time steps. They can be computed by raising the one-step transition matrix to a power corresponding to the number of steps, thereby applying the Chapman-Kolmogorov equations, which ensure consistency of the overall transition process.

Poisson Distribution

The Poisson distribution is a probability distribution that is often used to model the number of events occurring within a fixed interval of time or space when these events happen independently and with a known constant mean rate. It is characterized by its parameter (mean), and is useful in settings where events occur rarely or randomly.

Markov Chains

A Markov chain is a stochastic process that undergoes transitions between a set of discrete states with the probability of each future state depending only on the current state, not on the sequence of events that preceded it. This memoryless property makes Markov chains a powerful tool for modeling random processes in various fields.

Transition Matrix

The transition matrix in a Markov chain is a square matrix that describes the probabilities of transitioning from any given state to every other state in one step. Each entry in the matrix represents the probability of moving from one state to another in a single time interval, and the rows of the matrix typically sum up to one.

Transcript

00:01 In this problem, we are told that the number of visits that a person makes to an auto repair shop in a given year follows a poisson distribution, where the poisson parameter is based on the number of visits that they made in the previous year.

00:19 So, for example, if they made zero visits in the previous year, then the number of visits in the next year is a poisson distribution with this parameter.

00:32 And so for part a, we're asked to construct the one -step transition matrix for the number of repair shop visits by a randomly selected customer in the nth observed year.

00:44 Now, recall that poisson distribution for x and parameter mu is given by this equation.

01:08 And so the probability that x sub n plus 1 equals x given x sub n plus n equals x given x sub n, was i visits is a possible with this probability distribution.

01:36 So we could say that the one -step probability of going from zero visits in the previous year to zero visits in the next year is equal to the probability of zero given this rate.

01:56 And so that rate muz sub -zero is this value, and so that's equal to e to the minus 1 .933.

02:10 Approximately times 1 .939 to the exponent 0 over 0 factorial, which comes out to approximately 0 .144.

02:31 And we continue with those calculations for all single -step transitions.

02:36 Now because there are five possible states, we know that the transition matrix is going to be 5 by 5.

02:42 So you would save a lot of time by calculating these probabilities using software.

02:49 So that's what i did, and here's the one -step transition matrix that resulted.

02:58 And so this answer is part a.

03:04 So for part b, we are asked that if a customer makes two visits this year, what is the probability that customer makes two visits in the next year, and two visits in the year after that? so it helps to index these just to make sure we make less mistakes.

03:23 So we're looking for the one -step probability that we go from two visits to two visits, and then we again go from two visits to two visits.

03:47 So this represents two visits in the next year, and this represents two visits in the year after that.

03:56 And they are both one -step probabilities, one -step transitions, because the first time we're just going from this year into the next year...

Please give Ace some feedback