5 Consider the success run chain in Example 8.2.16. Suppose that the chain has been

running for a while and is currently in state 10. (a) What is the expected number of

steps until the chain is back at state 10? (b) What is the expected number of times the

chain visits state 9 before it is back at 10?

16 Consider a version of the success run chain in Example 8.2.16 where we disregard

sequences of consecutive tails, in the sense that for example T, T T, T T T, and so on,

all simply count as T. Describe this as a Markov chain and examine it in terms of

irreducibility, recurrence, and periodicity. Find the stationary distribution and compare

with Example 8.2.16. Is it the limit distribution?

17 Reversibility. Consider an ergodic Markov chain, observed at a late timepoint n. If

we look at the chain backward, we have the backward transition probability qij =

P(Xn?1 = j|Xn = i). (a) Express qij in terms of the forward transition probabilities

and the stationary distribution ?. (b)If the forward and backward transition probabilities

are equal, the chain is called reversible. Show that this occurs if and only if ?ipij =

?jpji for all states i, j (this identity is usually taken as the definition of reversibility).

(c) Show that if a probability distribution ? satisfies the equation ?ipij = ?jpji for all

i, j, then ? is stationary.

18 The intuition behind reversibility is that if we are given a sequence of consecutive states

under stationary conditions, there is no way to decide whether the states are given in

forward or backward time. Consider the ON/OFF system in Example 8.2.4; use the

definition in the previous problem to show that it is reversible and explain intuitively.

19 For which values of p is the following matrix the transition matrix of a reversible Markov

chain? Explain intuitively.

P =

0 p 1 ? p

1 ? p 0 p

p 1 ? p 0

!

20 Ehrenfest model of diffusion. Consider two containers containing a total of N gas

molecules, connected by a narrow aperture. Each time unit, one of the N molecules is

chosen at random to pass through the aperture from one container to the other. Let Xn

be the number of molecules in the first container. (a) Find the transition probabilities

for the Markov chain {Xn}. (b) Argue intuitively why the chain is reversible and why

the stationary distribution is a certain binomial distribution. Then use Problem 17 to

show that it is indeed the stationary distribution. (c) Is the stationary distribution also

the limit distribution?

21 Consider an irreducible and positive recurrent Markov chain with stationary distribution

? and let g : S ? R be a real-valued function on the state space. It can be shown that

1

n

Xn

k=1

g(Xk)

P?

X

j?S

g(j)?j

for any initial distribution, where we recall convergence in probability from Section 4.2.

This result is reminiscent of the law of large numbers, but the summands are not i.i.d.

We have mentioned that the interpretation of the stationary distribution is the long-term

proportion of time spent in each state. Show how a particular choice of the function g

above gives this interpretation (note that we do not assume aperiodicity.

1. WM 01 N10\" What is the difference between descriptive and inferential statistics? . Dene the analytical notion of metrics. . Suppose the speed limits in thirteen countries in miles per hour are as follows: Country Highway Miles per Hour 1. Italy 87 2. France 82 3. Hungary 75 4. Belgium 75 5. Portugal 75 6. Great Britain 70 7. Spain 62 8. Denmark 62 9. Netherlands 62 1 0. Greece 62 1 1. Japan 62 12. Norway 56 1 3 . Turkey 56 What is the mean, median, and mode for these data? Feel ee to use your computer (statistical software or spreadsheet) to get the answer.Which is the best measure of central tendency for these observations? Prepare a frequency distribution for the data in Question 3. . Why is the standard deviation rather than the average deviation typically used? . Calculate the standard deviation for the data in Question 3. . Draw three distributions that have the same mean value but dif- ferent standard deviation values. Draw three distributions that have the same standard deviation value but different mean values. . A smartphone manufacturer surveyed 100 retail phone outlets in each of the rm's sales regions.An analyst noticed that in the South Atlantic region the average retail price was $165 (mean) and the standard deviation was 330. However, in the Mid- Atlantic region the mean price was $170, with a standard devia- tion of $15.What do these statistics tell us about these two sales regions? (0) /2. 3. The Catalan number On gives the number possible non-intersecting diagonals in a a convex polygonProblem 4: In a basic repetition code, every transmitted bit is repeated 3 times. that is: to transmit a III. the codeword i} is sent and to transmit a 1. the codeword 111 is sent The receiver takes the three received bits and decides which hit was sent via a majorityr vote of the three hits. Assume that the coded bits are transmitted at the rate of 3l'v'lbps and that for each bit, a transmission error may occur randomly and independently with probability of error p = l'g. [a]: Determine the following parameters of the code: - Bode rate; - Minimum distance; - Number of correctable errors; - Number of detects ble errors. {b} For n = 11.1.2.3. nd the probability of making to errors when transmitting a single 3-bit codeword. [c] Briefly explain the rationale behind the error correction mechanism (i.e.. the voting scheme]. For simplicity+ consider the transmission of a D input bit. {d} Find the probability that the receiver makes an incorrect decision? (10 points) 13. This question is on graphs and trees. What is the definition of a graph as given in Section 10.1? b) Provide an example of an application of a graph. c) Provide an example of an application of a tree. (See Section 1 1.1 of the text.) d) Graphs and trees have a plethora of applications in computer science and mathematics. As time permits, read Chapter 10 and 1 1. In particular, read about tree traversal. I encourage you to continue to read this material over the break. If you are studying computer science, this topic will reappear in your data structures and algorithm courses. There is nothing for you to turn in for this part