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)

2. (10 pts) Picking stickers. Envelope 1 contains 51 Stat 134 stickers and d1 Stat 140 stickers. Envelope 2 contains 52 Stat 134 stickers and d2 Stat 140 stickers. Your friend chooses one envelope uniformly at random and hands it to you. You pick one sticker out uniformly at random and it is a Stat 134 sticker. Without replacing the sticker you just picked, what is the chance that the next sticker you pick (also uniformly at random) from the same envelope is a Stat 140 sticker? (1 point) In a large population the probability of someone being stricken with the euphoric mental state "StatsOnMyMind-itis" is 0.122. The probability of a person with StatsOnMyMind-itis of getting into a stats-related conversation is 0.88. The probability of a person that doesn't have StatsOnMyMind-itis of getting into a stats-related conversation is 0.839. (a) What is the probability of having StatsOnMyMind-itis and ending up in a stats-related conversation? (b) What is the probability of having StatsOnMyMind-itis and not ending up in a stats-related conversation? (c) What is the probability of not having StatsOnMyMind-itis and ending up in a stats-related conversation? (d) What is the probability of not having StatsOnMyMind-itis and not ending up in a stats-related conversation? (e) What is the probability of having StatsOnMyMind-itis given that you just ended up in a stats-related conversation? (f) What is the probability of not having StatsOnMyMind-itis given that you just ended up in a stats-related conversation? (g) What is the probability of having StatsOnMyMind-itis given that you have not ended up in a stats-related conversation? (h) What is the probability of not having StatsOnMyMind-itis given that you have not ended up in a stats-related conversation?A) Select all of the following variables for which a pie chart would be a valid visual Group of answer choices - final letter grades for students in STAT 2300 - the gender of students enrolled in STAT 2300 - the number of Economics majors taking STAT 2300 - final scores for students in STAT 2300 B) Select all of the following variables that could be displayed using a Pareto chart. Group of answer choices - the eye color of a student in STAT 2300 - the age of a student in STAT 2300 - the anticipated year of graduation for a student in STAT 2300 - the classification (freshman, sophomore, etc.) of a student in STAT 2300(h) Set plh=rgb_color([1 0 1]). (i) Set pli=rgb_color([0 0 0]). Problem 2: Write a function my stat.m to compute statistics of a vector. The function should have the following header: function [stat] = my stat (vect) where the input vect is a 3-element row vector. The function produces the statistics in the output vector stat in the following manner : . The first element of stat has the median value of vect. . The second element of stat has the minimum value of vect. . The third (last) element of stat has the maximum value of vect. . If any two elements in vect are equal, then the two equal elements are listed first in stat. . If all three elements in vect are equal, then stat is equal to vect. For example, for vect = [1 3 2], stat = [2 1 3]. For vect = [1 3 3], stat = [3 3 1]. Remember to give the function a description. You can use any type of if statements as well as the built-in functions: max, min, and median. (a) Set p2a=evalc('help my stat'). (b) Set p2b=my stat([10.5 12.6 10.9]). (c) Set p2c=my stat ([130 313 420]). (d) Set p2d=my stat([12 12 9]). (e) Set p2e=my stat([8 7 7]). (f) Set p2f=my stat ([6 9 6 ]). (g) Set p2g=my stat([5 5 5])