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theory of probability

Questions and Answers of Theory Of Probability

In the generalized random-walk problem of section 8 put [in analogy with (8.1)] P Pa+Pa+1-2 +Pa+2-2 +, and let dn be the probability that the game lasts for exactly n steps. Show that for n 1 a-1
In the random walk with absorbing barriers at the points 0 and a and with initial position z, let w,n(x) be the probability that the nth step takes the particle to the position . Find the difference
Continuation. Modify the boundary conditions for the case of two reflecting barriers (i.e., elastic barriers with 8 = 1).
A symmetric random walk (p-q) has possible positions 1, 2, ., a1. There is an absorbing barrier at the origin and a reflecting barrier at the other end. Find the generating function for the waiting
An alternative form for the first-passage probabilities. In the explicit formula (5.7) for the ruin probabilities let a co. Show that the result is . cos-1x sin x sin xz. dx. Consequently, this
Continuation: First passages in diffusion. Show that the passage to the limit described in section 6 leads from the last formula to the expression 2TD13 e-(2+1)/(2D) for the probability density for
Method of images. 14 Let p = q = In a random walk in (0, co) with an absorbing barrier at the origin and initial position at > 0, let uz,n(x) be the probability that the nth step takes the particle
Continuation. If the origin is a reflecting barrier, then U,n(x) = 0x2,n+0x+2-1,"
Continuation. If the random walk is restricted to (0,a) and both barriers are absorbing, then (9.1) - Uz,n(x) = {Vk-2kan - Vz+z2ka,n}, the summation extending over all k, positive or negative (only
Distribution of maxima. In a symmetric unrestricted random walk starting at the origin let Mn be the maximum abscissa of the particle in n steps. Using problem 15, show that (9.2) V(s) = = P{M} =
Let V(s) Uns" (cf. the note preceding problem 15). Prove that Vo(a)(s) when x
In a random walk in (0, ) with an absorbing barrier at the origin and initial position at z, let u,n(x) be the probability that the nth step takes the particle to the position x, and let (9.3) 00
Alternative formula for the probability of ruin (5.7). Expanding (4.11) into a geometric series, prove that 00 ..n = Pika W+2ka.n - 00 \ka-z W2ka-2,1 k=1 where w denotes the first-passage probability
If the passage to the limit of section 6 is applied to the expression for Un given in the preceding problem, show that the probability density of the absorption time equals 15 1 V2TD13 00 (c+2)/(2D)
Renewal method for the ruin problem. 16 In the random walk with two absorbing barriers let un and un be, respectively, the probabilities of absorption at the left and the right barriers. By a proper
Let u(x) be the probability that the particle, starting from z, will at the nth step be at a without having previously touched the absorbing barriers. Using the notations of problem 23, show that for
Continuation. The generating function U(s; x) of the preceding problem can be obtained by putting U(s; x) = V(s) - A(s) - B(s) and determining the constants so that the boundary conditions U(s; x) =
Prove the formula Vz,n = (2m)12^p(n+2)/2q(n2)/2 L Cost cos tx dt Vx(s) = (2)-1 L COS tx dt. 1-2 pq s cos f by showing that the appropriate difference equation is satisfied. Conclude that
In a three-dimensional symmetric random walk the particle has prob- ability one to pass infinitely often through any particular line x = m, y = n. (Hint: Cf. problem 5.)
In a two-dimensional symmetric random walk starting at the origin the probability that the nth step takes the particle to (x, y) is (27)-22-n *[*[* (cos x + cos f)" + cos xx cos y dad. Verify this
In a two-dimensional symmetric random walk let D = x + y be the square of the distance of the particle from the origin at time n. Prove E(D) n. [Hint: Calculate E(D%A1-1-D).]
In a symmetric random walk in d dimensions the particle has probability 1 to return infinitely often to a position already previously occupied. Hint: At each step the probability of moving to a new
Show that the method described in section 8 works also for the generating function U(s) of the waiting time for ruin.
In a sequence of Bernoulli trials we say that at time n the state is observed if the trials number n - 1 and n resulted in SS. Similarly E, E3, E4 stand for SF, FS, FF. Find the matrix P and all its
Classify the states for the four chains whose matrices P have the rows given below. Find in each case P2 and the asymptotic behavior of p. (a) (0, 1), (1, 0, 1), (1, 1, 0); (b) (0, 0, 0, 1), (0, 0,
We consider throws of a true die and agree to say that at epoch n the system is in state E, if j is the highest number appearing in the first n throws. Find the matrix p" and verify that (3.3) holds.
In example (2.j) find the (absorption) probabilities x and y that, starting from Ex, the system will end in E or Es, respectively (k = 2, 3, 4, 6). (Do this problem from the basic definitions without
Treat example I, (5.b) as a Markov chain. Calculate the probability of winning for each player.
Let Eo be absorbing (that is, put Poo = 1). For >0 let Pi; = P and P.-19, where p + q = 1.Find the probability f(n) that absorption at Eo takes place exactly at the nth step. Find also the
The first row of the matrix P is given by 0, 1, ... For >0 we have (as in the preceding problem) P = p and P-1 = q. Find the distribution of the recurrence time for Eo.
For = 0, 1,... let Pi, 5+2 = v; and Pjo = 1,. Discuss the character of the states.
Two reflecting barriers. A chain with states 1, 2,..., p has a matrix whose first and last rows are (q, p, 0,..., 0) and (0,..., 0, q, p). In all other rows Pk,k+1 P.Pk.k-19. Find the stationary
Generalize the Bernoulli-Laplace model of diffusion [example (2.f)] by assuming that there are b p black particles and w = 2p-6 white ones. The number of particles in each container remains = p.
A chain with states Eo, E,... has transition probabilities = Pik e2 v=0 2k-v (k-v)! where the terms in the sum should be replaced by zero if > k. Show that pela. (2/9) k!
Ehrenfest model. In example (2.e) let there initially be j molecules in the first container, and let X(") = 2k -a if at the nth step the system is in state k (so that Xn) is the difference of the
Treat the counter problem, example XIII, (1.g), as a Markov chain.
Plane random walk with reflecting barriers. Consider a symmetric random walk in a bounded region of the plane. The boundary is reflecting in the sense that, whenever in a unrestricted random walk the
Repeated averaging. Let {x1, x2,...} be a bounded sequence of numbers and P the matrix of an ergodic chain. Prove that por / Eu,x,. Show that the repeated averaging procedure of example XIII, (10.c)
In the theory of waiting lines we ecounter the chain matrix Po P1 P2 P3 Po P1 P2 P3 0 Po P1 P2 0 0 Po P1 where {P} is a probability distribution. Using generating functions, discuss the character of
Waiting time to absorption. For transient E, let Y, be the time when the system for the first time passes into a persistent state. Assuming that the probability of staying forever in transient states
If the number of states is a 25.Moving averages. Let (Y) be a sequence of mutually independent random variables, each assuming the values 1 with probability
Put X(n) = (Y+Y+1)/2. Find the transition probabilities - Pix(m, n) = P{X(n) = k|x(m) = j}, where m 28.Let N be a Poisson variable with expectation
Consider N inde- pendent Markov processes starting at Eo and having the same matrix P. Denote by Zn) the number among them after n steps are found in state Ex- Show that Z) has a Poisson distribution
Among the digits 1, 2, 3, 4, 5 first one is chosen, and then a second selection is made among the remaining four digits. Assume that all twenty possible re- sults have the same probability. Find the
In the sample space of example (2.a) attach equal probabilities to all 27 points. Using the notation of example (4.d), verify formula (7.4) for the two events A = S and A = S2. How many points does
Consider the 24 possible arrangements (permutations) of the symbols 1234 and attach to each probability 24.Let A, be the event that the digit i appears at its natural place (where i 1, 2, 3, 4).
A coin is tossed until for the first time the same result appears twice in succession. To every possible outcome requiring n tosses attribute probability 1/2 Describe the sample space. Find the
In the sample space of example (5.b) let us attribute to each point of (*) containing exactly k letters probability 1/2. (In other words, aa and bb carry probability, acb has probability, etc.) (a)
Modify example (5.6) to take account of the possibility of ties at the individual games. Describe the appropriate sample space. How would you define probabilities?
In problem 3 show that AA2A3 A4 and AA2A = A.
Using the notations of example (4.d) show that (a) SS2D3 = 0; (b) SD2E3; (c) E3 - DS > S2D1.
Two dice are thrown. Let A be the event that the sum of the faces is odd, B the event of at least one ace. Describe the events AB, AUB, AB'. Find their probabilities assuming that all 36 sample
In example (2.g), discuss the meaning of the following events: (a) ABC, (b) A - AB, = (c) AB'C.
In example (2.g), verify that AC' = B.
Bridge (cf. footnote 1). For k 1, 2, 3, 4 let N be the event that North has at least k aces. Let S, E, W be the analogous events for South, East, West. Discuss the number x of aces in West's
In the preceding problem verify that (a) S3 = S, (d) NS = W,
Verify the following relations.4 (a) (AUB)'= A'B'. (c) AAA UA - A. - (e) (AUB) AB = AB' A'B. (g) (AUB)C - AC U BC.
Find simple expressions for - (b) (AB) B = A - AB = AB'. (d) (A-AB) UB = AUB. (f) A'B' (AB)'. - (a) (AUB)(A U B'), (b) (AUB)(A' UB)(AUB), (c) (AUB)(BUC).
State which of the following relations are correct and which incorrect: (a) (AUB) C=AU (B-C). (b) ABC = AB(CUB). (c) AUBUC=AU (B-AB) U (C-AC). = (d) AUB (A-AB) U B. (e) AB UBC UCA > ABC. (f) (ABBC
Let A, B, C be three arbitrary events. Find expressions for the events that of A, B, C: (a) Only A occurs. (c) All three events occur. (e) At least two occur. (g) Two and no more occur. (i) Not more
The union AUB of two events can be expressed as the union of two mutually exclusive events, thus: AUB = AU (B-AB). Express in a similar way the union of three events A, B, C.
How many different sets of initials can be formed if every person has one surname and (a) exactly two given names, (b) at most two given names, (c) at most three given names?
Letters in the Morse code are formed by a succession of dashes and dots with repetitions permitted. How many letters is it possible to form with ten symbols or less?
Each domino piece is marked by two numbers. The pieces are symmetrical so that the number-pair is not ordered. How many different pieces can be made using the numbers 1, 2, ..., n?
The numbers 1, 2, . . ., n are arranged in random order. Find the proba- bility that the digits (a) 1 and 2, (b) 1, 2, and 3, appear as neighbors in the order named.
A throws six dice and wins if he scores at least one ace. B throws twelve dice and wins if he scores at least two aces. Who has the greater probability to win?18 Hint: Calculate the probabilities to
(a) Find the probability that among three random digits there appear exactly 1, 2, or 3 different ones. (b) Do the same for four random digits.
Find the probabilities p, that in a sample of r random digits no two are equal. Estimate the numerical value of P10, using Stirling's formula.
What is the probability that among k random digits (a) 0 does not appear; (b) 1 does not appear; (c) neither 0 nor 1 appears; (d) at least one of the two digits 0 and 1 does not appear? Let A and B
At a parking lot there are twelve places arranged in a row. A man ob- served that there were eight cars parked, and that the four empty places were adjacent to each other (formed one run). Given that
A man is given n keys of which only one fits his door. He tries them successively (sampling without replacement). This procedure may require 1, 2,...,n trials. Show that each of these n outcomes has
Suppose that each of n sticks is broken into one long and one short part. The 2n parts are arranged into n pairs from which new sticks are formed. Find the probability (a) that the parts will be
Testing a statistical hypothesis. A Cornell professor got a ticket twelve times for illegal overnight parking. All twelve tickets were given either Tuesdays or Thursdays. Find the probability of this
Continuation. Of twelve police tickets none was given on Sunday. Is this evidence that no tickets are given on Sundays?
A box contains ninety good and ten defective screws. If ten screws are used, what is the probability that none is defective?
From the population of five symbolsa, b,c, d,e, a sample of size 25 is taken. Find the probability that the sample will contain five symbols of each kind. Check the result in tables of random
If n men, among whom are A and B, stand in a row, what is the probability that there will be exactly r men between A and B? If they stand in a ring instead of in a row, show that the probability is
What is the probability that two throws with three dice each will show the same configuration if (a) the dice are distinguishable, (b) they are not?
Show that it is more probable to get at least one ace with four dice than at least one double ace in 24 throws of two dice. The answer is known as de Mr's paradox.21
From a population of n elements a sample of size r is taken. Find the probability that none of N prescribed elements will be included in the sample, assuming the sampling to be (a) without, (b) with
Spread of rumors. In a town of n + 1 inhabitants, a person tells a rumor to a second person, who in turn repeats it to a third person, etc. At each step the recipient of the rumor is chosen at random
Chain letters. In a population of n + 1 people a man, the "progenitor," sends out letters to two distinct persons, the "first generation." These repeat the performance and, generally, for each letter
A family problem. In a certain family four girls take turns at washing dishes. Out of a total of four breakages, three were caused by the youngest girl, and she was thereafter called clumsy. Was she
What is the probability that (a) the birthdays of twelve people will fall in twelve different calendar months (assume equal probabilities for the twelve months), (b) the birthdays of six people will
Given thirty people, find the probability that among the twelve months there are six containing two birthdays and six containing three.
A closet contains n pairs of shoes. If 2r shoes are chosen at random (with 2r
Leta, b,c, d be four non-negative integers such that a+b+c+d= 13.Find the probability p(a,b, c,d) that in a bridge game the players North, East, South, West havea, b,c, d spades, respectively.
Using the result of problem 32, find the probability that some player receivesa, anotherb, a thirdc, and the last d spades if (a) a = 5, b = 4, c = 3, d = 1; (b) a = b = c = 4, d = 1; (c) a = b = 4,
Leta, b,c, d be integers with a+b+c+d = 13.Find the probability q(a,b, c,d) that a hand at bridge will consist of a spades, b hearts, c dia- monds, and d clubs and show that the problem does not
Distribution of aces among r bridge cards. Calculate the probabilities Po(r), Pi(r), P4(r) that among r bridge cards drawn at random there are 0,1,..., 4 aces, respectively. Verify that po(r) =
Continuation: waiting times. If the cards are drawn one by one, find the probabilities f(r), fa(r) that the first,..., fourth ace turns up at the rth trial. Guess at the medians of the waiting times
Find the probability that each of two hands contains exactly k aces if the two hands are composed of r bridge cards each, and are drawn (a) from the same deck, (b) from two decks. Show that when r =
Misprints. Each page of a book contains N symbols, possibly mis- prints. The book contains n = 500 pages and r = 50 misprints. Show that32. Leta, b,c, d be four non-negative integers such that
If r indistinguishable things of one kind and r indistinguishable things of a second kind are placed into n cells, find the number of distinguishable arrangements.
If dice and 2 coins are thrown, how many results can be distin- guished?
In how many different distinguishable ways can r white, r2 black, and 13 red balls be arranged?
Find the probability that in a random arrangement of 52 bridge cards no two aces are adjacent.
Elevator. In the example (3.c) the elevator starts with seven passengers and stops at ten floors. The various arrangements of discharge may be denoted by symbols like (3, 2, 2), to be interpreted as
Birthdays. Find the probabilities for the various configurations of the birthdays of 22 people.
Find the probability for a poker hand to be a (a) royal flush (ten, jack, queen, king, ace in a single suit); (b) four of a kind (four cards of equal face values); (c) full house (one pair and one

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