代写STATS 325/721、代做R编程设计、R设计代写、代做diagram

日期：2019-09-01 10:58

STATS 325/721 Assignment 2

1. Consider a Markov chain X = (Xn : n = 0, 1, 2, . . .) on the set of vertices {A, B, C, D, E}

of the 3D object with six triangular faces represented in the following diagram:

Let Pi and Ei represent probability and expectation, respectively, conditional on

X0 = i. The first hitting time of state i is defined by

Ti:= inf{n > 0 : Xn = i}.

(a) Prove that

PB(TA < TD) = 37,

where you should justify your steps. [4]

(b) Calculate PE(TB < TA; TB < TD), and deduce PC(TB < TD < TA). [4]

(c) Find the average time, EA(TB), it takes to reach state B starting initially in

state A. [4]

(d) Deduce the average time to return to state B starting from state B. Deduce

the long term proportion of time spent in each state. [

?3]

2. Consider the Markov chain X = (Xn)n∈N with state space I = {A, B, C, D, E, F}

and one step transition probabilities given in the following diagram:

(a) Decompose the state space into its communicating classes and state the period

of each class. Hence, identify the set of transient states T and a communicating

class of recurrent states R. [3]

Due noon, Monday, September 2nd 2019 (Science SRC)

STATS 325/721 Assignment 2

(b) Write down the one-step transition matrix P for the discrete parameter Markov

chain Y with state space R, that is, the restriction of the Markov chain X to

the recurrent class R ? I. [3]

(c) What conditions does an invariant probability mass function π for a discrete

time Markov chain satisfy? Find π for the Markov chain Y . [3]

(d) Stating any general results that you appeal to, deduce the following:

i. Y is positive recurrent, [1]

ii. the distribution for the position of Y after the chain has been running for

a very long time, [1]

iii. the long-term proportion of time spent in each of the states, [1]

iv. the average time to return to each state EiTi

, [1]

v. the average number of visits made to A before returning to the starting

position at C. [

?2]

3. Consider the random walk W = (Wn)n≥0 with state space Z such that

Wn := W0 + X1 + · · · + Xn,

where X1, X2, . . . are independent, identically distributed random variables with

P(Xn = ?1) = 25, P(Xn = 1) = 15, P(Xn = 2) = 25.(a) For k ≥ 1, let xk be the probability that the random walk ever visits the origin

given that it starts at position k, that is,

xk := Pk(hit 0) := P(Wn = 0 for some n ≥ 0 | W0 = k).

i. By splitting according to the first move, show that

and explain why xk = (x1)

k

for k ≥ 1. [5]

ii. Show that Pk(hit 0) = 2?k

for k ≥ 1. [5]

(b) For k ≥ ?1, let yk be the probability that the random walk ever visits k given

that it starts at 0, that is,

yk := P0(hit k) := P(Wn = k for some n ≥ 0 | W0 = 0).

i. Write down the values of y?1 and y0. [2]

ii. For k ≥ 1, briefly explain why

iii. Find all solutions to (?) of the form yk ∝ mk and write down the general

solution of the recurrence relation (?). Deduce P0(hit k) for k ≥ ?1. [

?4]

iv. If the random walk starts at the origin and n > 0 is a very large integer, deduce

that the probability that position n is never visited is approximately

1/6. [

?1]

Stats325: mark is out of 40, including max 5 bonus from ?

starred questions

Stats721: mark is out of 50, please attempt all questions

Due noon, Monday, September 2nd 2019 (Science SRC)