6. Suppose that a random walk S, (in some graph/environment, such as Zd, butnot necessarily) is recurrent when started from position x. That is,P(n ≥ 1, Sn=x|So = x) = 1.Moreover, suppose that there is positive probability that the walk will eventu-ally visit some other site y, when started from x. That is,P(n ≥ 1, Sny|So = x) > 0.Argue that, in fact, the random walk will visit y infinitely often, when startedfrom x.Hint: Try a proof by contradiction, using the Law of Large Numbers.

Given that, Sn is a random walk is recurrent when started from position x. P∃n≥1, Sn=x|S0=x=1

Answered: 6. Suppose that a random walk S₂ (in…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

5. Let {S, n>0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X, for n ≥ 1, where X₁, i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1 -p for i>1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive integers. Define T = min{n: S₁ = -a or S₁₁=b}. Assume P(T<∞) = 1. Explain why one can use Doob's Optional Stopping Theorem to conclude that E[ZT] = 1. We can use the conclusion that: T is a stopping time with respect to the o-fields Fn, n ≥ 0. A) B) Define Z₁ = (q/p) Sn, n ≥ 0. {Zn, n ≥ 0} is a martingale with respect to the o-fields Fn, n ≥ 0.
C Clever | Portal HISD OnTrack (11) Meeting its itslearning quizizz.com/join/game/U2FsdGVkX1%252FUj5zpnyXi%252FCE5P O online.houstonisd.org bookmarks Zoom - Chrome We.. 2/10 Streak 1.5 +0.75x s9 X > 10 X < 6.5
A.2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter lambda. a) If X1, X2. ..., Xp are the times, in minutes, between Successive customers selected randomly, estimate the parameter of the distribution. b) The randomly selected 15 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, 1.8, 0.9, 1.5 and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c) In order to estimate the distribution parameter with 0.4 error and 4% risk, find the minimum sample size.
4.1 Let X have a geometric distribution, that is p(x) = q*-'p x = 1,2,3, .. Where X has the memoryless property given by: P(X > k + b|X > k) = P(X > b). Given that you have already tossed a balanced coin 5 times and obtained Zero heads, what is the probability that you must toss it at least 3 more Times to obtain the first head?
2. Suppose the defective rate at a particular factory is 1%. Suppose 50 parts were selected from the daily output of parts. Let X denote the number of defective parts in the sample. a) Find the probability that the sample contains exactly 2 defective parts. b) Use Poisson approximation to find the probability that the sample contains exactly 2 defective parts. c) Find the probability that the sample contains at most 1 defective part. d) Use Poisson approximation to find the probability that the sample contains at most 1defective part.
Consider two types of call arrivals to a cell in a mobile telephone network: new calls that originate in a cell and calls that are handed over from neighboring cells. It is desirable to give preference to handover calls over new calls. For this reason, some of the channels in the cell are reserved for handover calls, while the rest of the channels are available to both types of calls. = = Suppose that all calls arrive in a cell according to a Poisson process with rate Anc 125 calls per hour for new calls, and λho 50 calls per hour for handover calls. The cell has a capacity of 10 channels. Each call occupies 1 channel and the call time has an exponential distribution with mean two minutes. Suppose that no channels are reserved for handover calls. (a) Calculate the blocking probability in the cell and find the average number of channels used. (b) When a call has to wait, find its average waiting time.
b) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?
Pls help
If a path of length 10 is randomly selected between point O(0,0) and point A(4,6), and movement is restricted to only travel along line segments connecting points with integer coordinates and parallel to the coordinate axes, calculate the probability that the path will intersect point B(1,2).
4. The distribution function of a random variable W is given by: Oforw 2 a. Find P (1 < w< 3). Enter probability as a reduced fraction. prob = %3D b. Find f(w). Enter coefficients as reduced fractions and express powers with ^. f(w) 3= for -2< w < 2 %3D

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS