Applied recurrence modeling

5 In this question, give probabilities correct to 4 decimal places.
Alpha and Delta are two companies which compete for the ownership of insurance bonds. Boyles and Cayleys are companies which trade in these bonds. When a new bond becomes available, it is first acquired by either Boyles or Cayleys. After a certain amount of trading it is eventually owned by either Alpha or Delta. Change of ownership always takes place overnight, so that on any particular day the bond is owned by one of the four companies. The trading process is modelled as a Markov chain with four states, as follows. On the first day, the bond is owned by Boyles or Cayleys, with probabilities \(0.4,0.6\) respectively.
If the bond is owned by Boyles, then on the next day it could be owned by Alpha, Boyles or Cayleys, with probabilities \(0.07,0.8,0.13\) respectively. If the bond is owned by Cayleys, then on the next day it could be owned by Boyles, Cayleys or Delta, with probabilities \(0.15,0.75,0.1\) respectively. If the bond is owned by Alpha or Delta, then no further trading takes place, so on the next day it is owned by the same company.

1. Write down the transition matrix \(\mathbf { P }\).
2. Explain what is meant by an absorbing state of a Markov chain. Identify any absorbing states in this situation.
3. Find the probability that the bond is owned by Boyles on the 10th day.
4. Find the probability that on the 14th day the bond is owned by the same company as on the 10th day.
5. Find the day on which the probability that the bond is owned by Alpha or Delta exceeds 0.9 for the first time.
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
7. Find the probability that the bond is eventually owned by Alpha. The probabilities that Boyles and Cayleys own the bond on the first day are changed (but all the transition probabilities remain the same as before). The bond is now equally likely to be owned by Alpha or Delta at the end of the trading process.
8. Find the new probabilities for the ownership of the bond on the first day.

5. \begin{figure}[h] \end{figure} Figure 1 shows the first three stages of a pattern that is created by a recursive process.
The process starts with a square and proceeds as follows

• each square is replaced by 5 smaller squares each \(\frac { 1 } { 9 }\) th the size of the square being replaced
• the 5 smaller squares are the ones in each corner and the one in the centre
• once each of the squares has been replaced, the square immediately to the right and above the centre square of the pattern is then removed

Let \(u _ { n }\) be the number of squares in the pattern in stage \(n\), where stage 1 is the original square.

1. Explain why \(u _ { n }\) satisfies the recurrence system $$u _ { 1 } = 1 \quad u _ { n + 1 } = 5 u _ { n } - 1 \quad ( n = 1,2,3 , \ldots )$$
2. Solve this recurrence system. Given that the initial square has area 25
3. determine the total area of all the squares in stage 8 of the pattern, giving your answer to 2 significant figures.

2. In two-dimensional space, lines divide a plane into a number of different regions. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} It is known that:

• One line divides a plane into 2 regions, as shown in Figure 1
• Two lines divide a plane into a maximum of 4 regions, as shown in Figure 2
• Three lines divide a plane into a maximum of 7 regions, as shown in Figure 3
• Four lines divide a plane into a maximum of 11 regions, as shown in Figure 4
  1. Complete the table in the answer book to show the maximum number of regions when five, six and seven lines divide a plane.
  2. Find, in terms of \(\mathrm { u } _ { \mathrm { n } }\), the recurrence relation for \(\mathrm { u } _ { \mathrm { n } + 1 }\), the maximum number of regions when a plane is divided by ( \(n + 1\) ) lines where \(n \geqslant 1\)
  3. 1. Solve the recurrence relation for \(u _ { n }\)
     2. Hence determine the maximum number of regions created when 200 lines divide a plane.

1. The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus \(k\) times the number of visits received in the month before that, where \(k\) is a positive constant.

Given that \(V _ { n }\) is the number of visits to the website in month \(n\),

1. write down a general recurrence relation for \(V _ { n + 2 }\) in terms of \(V _ { n + 1 } , V _ { n }\) and \(k\). For a particular website you are given that
   • \(k = 0.24\)
   • In month 1 , there were 65 visits to the website.
   • In month 2 , there were 71 visits to the website.
   • Show that
   $$V _ { n } = 50 ( 1.2 ) ^ { n } - 25 ( - 0.2 ) ^ { n }$$ This model predicts that the number of visits to this website will exceed one million for the first time in month \(N\).
2. Find the value of \(N\).

3. \begin{figure}[h] \end{figure} There are three lily pads on a pond. A frog hops repeatedly from one lily pad to another.
The frog starts on lily pad A, as shown in Figure 1.
In a model, the frog hops from its position on one lily pad to either of the other two lily pads with equal probability. Let \(p _ { n }\) be the probability that the frog is on lily pad A after \(n\) hops.

1. Explain, with reference to the model, why \(p _ { 1 } = 0\) The probability \(p _ { n }\) satisfies the recurrence relation $$p _ { n + 1 } = \frac { 1 } { 2 } \left( 1 - p _ { n } \right) \quad n \geqslant 1 \quad \text { where } p _ { 1 } = 0$$
2. Prove by induction that, for \(n \geqslant 1\) $$p _ { n } = \frac { 2 } { 3 } \left( - \frac { 1 } { 2 } \right) ^ { n } + \frac { 1 } { 3 }$$
3. Use the result in part (b) to explain why, in the long term, the probability that the frog is on lily pad A is \(\frac { 1 } { 3 }\)

1. In a model for the number of subscribers to a new social media channel it is assumed that

• each week \(20 \%\) of the subscribers at the start of the week cancel their subscriptions
• between the start and end of week \(n\) the channel gains \(20 n\) new subscribers

Given that at the end of week 1 there were 25 subscribers,

1. explain why the number of subscribers at the end of week \(n , U _ { n }\), is modelled by the recurrence relation $$U _ { 1 } = 25 \quad U _ { n + 1 } = 0.8 U _ { n } + 20 ( n + 1 ) \quad n = 1,2,3 , \ldots$$
2. Prove by induction that for \(n \geqslant 1\) $$U _ { n } = 325 \left( \frac { 4 } { 5 } \right) ^ { n - 1 } + 100 n - 400$$ Given that 6 months after starting the channel there were approximately 1800 subscribers,
3. evaluate the model in the light of this information.

1. A staircase has \(n\) steps. A tourist moves from the bottom (step zero) to the top (step \(n\) ). At each move up the staircase she can go up either one step or two steps, and her overall climb up the staircase is a combination of such moves.

If \(u _ { n }\) is the number of ways that the tourist can climb up a staircase with \(n\) steps,

1. explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { n } = u _ { n - 1 } + u _ { n - 2 } , \text { with } u _ { 1 } = 1 \text { and } u _ { 2 } = 2$$
2. Find the number of ways in which she can climb up a staircase when there are eight steps. A staircase at a certain tourist attraction has 400 steps.
3. Show that the number of ways in which she could climb up to the top of this staircase is given by $$\frac { 1 } { \sqrt { 5 } } \left[ \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { 401 } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { 401 } \right]$$

7 In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, \(t\) years after their introduction, is denoted by \(N _ { t }\). The initial number of breeding pairs is given by \(N _ { 0 }\). An initial discrete population model is proposed for \(N _ { t }\). $$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$

1. (a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .
   (b) Show that \(N _ { t + 1 } - N _ { t } < 150 - N _ { t }\) when \(N _ { t }\) lies between 0 and 150 .
   (c) Hence determine the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when \(N _ { 0 } \in ( 0,150 )\). An alternative discrete population model is proposed for \(N _ { t }\). $$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
2. Given that \(N _ { 0 } = 8\), find the value of \(N _ { 4 }\) for each of the two models and give a reason why Model II may be considered more suitable.