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theory of probability

Questions and Answers of Theory Of Probability

1.10 For any set of numbers x1, ··· , xn and a monotone function h(·), show that the value of a that minimizesn i=1[h(xi)−h(a)]2 is given by a = h−1 n i=1 h(xi)/n. Find functions h that
1.9 (a) The median of any set of distinct real numbers x1,...,xn is defined to be the middle one of the ordered x’s when n is odd, and any value between the two middle ordered x’s when n is even.
1.8 If φ(a) = E|X − a| < ∞ for somea, show that φ(a) is minimized by any median of X. [Hint: If m0 ≤ m ≤ m1 (in the notation of Problem 1.7) and m1 m)] + 2 m
1.7 A median of X is any value m such that P(X ≤ m) ≥ 1/2 and P(X ≥ m) ≥ 1/2.(a) Show that this is equivalent to P(Xm) ≤ 1/2.(b) Show that the set of medians is always a closed interval m0
1.6 If X1,...,Xn are iid as C(a, b), the distribution of X¯ is again C(a, b). [Hint: Prove by induction, using Problem 5.]
1.5 Let Xi (i = 1, 2) be independently distributed according to the Cauchy densities C(ai, bi). Then, X1 + X2 is distributed as C(a1 + a2, b1 + b2). [Hint: Transform to new variables Y1 = X1 + X2, Y2
1.4 Let X and Y have common expectation θ, variances σ2 and τ 2, and correlation coefficient ρ. Determine the conditions on σ, τ , and ρ under which(a) var(X) < var[(X + Y )/2].(b) The value
1.3 In the preceding problem, minimize the variance of αiXi(αi = 1)(a) When the variance of Xi is σ2/αi (αi known).(b) When the Xi have common variance σ2 but are correlated with common
1.2 Let X1,...,Xn be uncorrelated random variables with common expectation θ and variance σ2. Then, among all linear estimators αiXi of θ satisfying αi = 1, the mean X¯ has the smallest
1.1 If (x1, y1),..., (xn, yn) are n points in the plane, determine the best fitting line y =α + βx in the least squares sense, that is, determine the values α and β that minimize[yi − (α +
In a Markov decision problem, another criterion often used, different than the expected average return per unit time, is that of the expected discounted return.In this criterion we choose a number
A Markov chain is said to be a tree process if(i) Pij > 0 whenever Pji > 0,(ii) for every pair of states i and j, i = j , there is a unique sequence of distinct states i = i0, i1, . . . ,
It follows from Theorem 4.2 that for a time reversible Markov chainIt turns out that if the state space is finite and Pij > 0 for all i, j , then the preceding is also a sufficient condition for
(a) Show that the limiting probabilities of the reversed Markov chain are the same as for the forward chain by showing that they satisfy the equations(b) Give an intuitive explanation for the result
For a branching process, calculate π0 when (a) Po, P2= (b) Po= , P = , P = - Po=, P =, P3.
In a branching process having X0 = 1 and μ>1, prove that π0 is the smallest positive number satisfying Eq. (4.20).Hint: Let π be any solution of π =∞j=0 πjPj . Show by mathematical
For the Markov chain with states 1, 2, 3, 4 whose transition probability matrix P is as specified below find fi3 and si3 for i = 1, 2, 3. P = 0.4 0.2 0.1 0.3 0.1 0.5 0.2 0.2 0.3 0.4 0.2 0.1 0 0 0 1
The following is the transition probability matrix of aMarkov chain with states 1, 2, 3, 4If X0 = 1 (a) find the probability that state 3 is entered before state 4;(b) find the mean number of
For the gambler’s ruin model of Section 4.5.1, let Mi denote the mean number of games that must be played until the gambler either goes broke or reaches a fortune of N, given that he starts with i,
In the gambler’s ruin problem of Section 4.5.1, suppose the gambler’s fortune is presently i, and suppose that we know that the gambler’s fortune will eventually reach N (before it goes to 0).
Consider the Ehrenfest urn model in which M molecules are distributed between two urns, and at each time point one of the molecules is chosen at random and is then removed from its urn and placed in
A Markov chain with states 1, . . . , 6 has transition probability matrix(a) Give the classes and tell which are recurrent and which are transient.(b) Find limn→∞Pn 1,2.(c) Find limn→∞Pn
Consider aMarkov chain with states 1, 2, 3 having transition probability matrix(a) If the chain is currently in state 1, find the probability that after two transitions it will be in state 2.(b)
Consider a Markov chain in steady state. Say that a k length run of zeroes ends at time m ifShow that the probability of this event is π0(P0,0)k−1(1 − P0,0)2, where π0 is the limiting
Let {Xn,n 0} denote an ergodicMarkov chain with limiting probabilities πi .Define the process {Yn,n 1} by Yn = (Xn−1,Xn). That is, Yn keeps track of the last two states of the original chain. Is
An individual possesses r umbrellas that he employs in going from his home to office, and vice versa. If he is at home (the office) at the beginning (end)of a day and it is raining, then he will take
Suppose that a population consists of a fixed number, say, m, of genes in any generation. Each gene is one of two possible genetic types. If exactly i (of the m) genes of any generation are of type
Each day, one of n possible elements is requested, the ith one with probability Pi, i 1,n1 Pi = 1. These elements are at all times arranged in an ordered list that is revised as follows: The element
Let A be a set of states, and let Ac be the remaining states.(a) What is the interpretation of ? iA jAc (b) What is the interpretation of ? iA < jA (c) Explain the identity ;; =; j iA jA
Consider a Markov chain with states equal to the nonnegative integers, and suppose its transition probabilities satisfy Pi,j = 0 if j ≤ i. Assume X0 = 0, and let ej be the probability that the
Capa plays either one or two chess games every day, with the number of games that she plays on successive days being a Markov chain with transition probabilitiesCapa wins each game with probability
Show that the stationary probabilities for the Markov chain having transition probabilities Pi,j are also the stationary probabilities for the Markov chain whose transition probabilities Qi,j are
The state of a process changes daily according to a two-state Markov chain. If the process is in state i during one day, then it is in state j the following day with probability Pi,j , whereEvery day
Let Yn be the sum of n independent rolls of a fair die. FindHint: Define an appropriate Markov chain and apply the results of Exercise 20. lim P{Yn is a multiple of 13} n
A transition probability matrix P is said to be doubly stochastic if the sum over each column equals one; that is,If such a chain is irreducible and consists of M + 1 states 0, 1, . . . , M, show
Specify the classes of the following Markov chains, and determine whether they are transient or recurrent: 1212 12 0 12 1212 P1 = 0 0 P2 = 0 0 0 1 00 0 1 12 12 0 0 1 0
Consider a Markov chain with transition probabilities qi,j , i, j ≥ 0. Let N0,k, k = 0, be the number of transitions, starting in state 0, until this Markov chain enters state k. Consider another
Let the transition probability matrix of a two-state Markov chain be given, as in Example 4.2, byShow by mathematical induction that P 1-P P = 1-p P
A Markov chain {Xn,n 0} with states 0, 1, 2, has the transition probability matrixIf P{X0 = 0} = P{X0 = 1} = 1 4 , find E[X3]. 1623 12 13 13 0 12 0 12
Recall that X is said to be a gamma random variable with parameters (α, λ) if its density is f (x) = λe−λx(λx)α−1/(α), x > 0(a) If Z is a standard normal random variable, show that Z2
Each new book donated to a library must be processed. Suppose that the time it takes a librarian to process a book has mean 10 minutes and standard deviation 3 minutes. If a librarian has 40 books
The standard deviation of a random variable is the positive square root of its variance. Letting σX and σY denote the standard deviations of the random variables X and Y, we define the correlation
Teams 1, 2, 3, 4 are all scheduled to play each of the other teams 10 times.Whenever team i plays team j , team i is the winner with probability Pi,j , wherea) Approximate the probability that team 1
With K(t) = log(E-etX.), show that K'(0) =E[X], K"(0) = Var(X)
Let φ(t1, . . . , tn) denote the joint moment generating function of X1, . . . , Xn.(a) Explain how the moment generating function of Xi,φXi (ti ), can be obtained from φ(t1, . . . , tn).(b) Show
Let X and Y be independent normal random variables, each having parametersμ and σ2. Show that X +Y is independent of X − Y.Hint: Find their joint moment generating function.
Show thatHint: Let Xn be Poisson with mean n. Use the central limit theorem to show that P{Xn ≤ n}→ 1 2 . lim en n IM= k=0 k! 12 *
If X is normally distributed with mean 1 and variance 4, use the tables to find P{2
Let X1,X2, . . . , X10 be independent Poisson random variables with mean 1.(a) Use the Markov inequality to get a bound on P{X1 +· · ·+X10 ≥ 15}.(b) Use the central limit theorem to approximate
Suppose that X is a random variable with mean 10 and variance 15. What can we say about P{5
Use Chebyshev’s inequality to prove the weak law of large numbers. Namely, if X1, X2, . . . are independent and identically distributed with mean μ and varianceσ2 then, for any ε >0, P
Consider Example 2.50. Find Cov(Xi, Xj ) in terms of the ars .
If X is Poisson with parameter λ, show that its Laplace transform is given by g(u)=E[eux]= e^(e"-1)
Consider n people and suppose that each of them has a birthday that is equally likely to be any of the 365 days of the year. Furthermore, assume that their birthdays are independent, and let A be the
Successive monthly sales are independent normal random variables with mean 100 and variance 100.(a) Find the probability that at least one of the next 5 months has sales above 115.(b) Find the
Show that the sum of independent identically distributed exponential random variables has a gamma distribution.
Calculate the moment generating function of a geometric random variable.
In deciding upon the appropriate premium to charge, insurance companies sometimes use the exponential principle, defined as follows. With X as the random amount that it will have to pay in claims,
Let X and W be the working and subsequent repair times of a certain machine.Let Y = X +W and suppose that the joint probability density of X and Y is fX,Y (x, y) = λ2e−λy , 0
Calculate the moment generating function of the uniform distribution on (0, 1).Obtain E[X] and Var[X] by differentiating.
Show that the random variables X1, . . . , Xn are independent if for each i =2, . . . , n, Xi is independent of X1, . . . , Xi−1.Hint: X1, . . . , Xn are independent if for any sets A1, . . . ,
The number of traffic accidents on successive days are independent Poisson random variables with mean 2.(a) Find the probability that 3 of the next 5 days have two accidents.(b) Find the probability
Show that Var(X) = 1 when X is the number of men who select their own hats in Example 2.30.
Let X denote the number of white balls selected when k balls are chosen at random from an urn containing n white and m black balls.(a) Compute P{X = i}.(b) Let, for i = 1, 2, . . . , k; j = 1, 2, . .
Let a1 < a2 < · · · < an denote a set of n numbers, and consider any permutation of these numbers. We say that there is an inversion of ai and aj in the permutation if i < j and aj precedes ai .
Let X1,X2, . . . be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time n if Xn > max(X1, . . . , Xn−1). That is, Xn is a record if it
Let X and Y be independent random variables with means μx and μy and variancesσ2 x and σ2 y . Show that Var(XY) = ++
Let X1,X2,X3, and X4 be independent continuous random variables with a common distribution function F and let(a) Argue that the value of p is the same for all continuous distribution functions F.(b)
An urn contains 2n balls, of which r are red. The balls are randomly removed in n successive pairs. Let X denote the number of pairs in which both balls are red.(a) Find E[X].(b) Find Var(X).
Suppose that X and Y are independent binomial random variables with parameters(n,p) and (m,p). Argue probabilistically (no computations necessary) that X + Y is binomial with parameters (n+ m,p).
There are n types of coupons. Each newly obtained coupon is, independently, type i with probability pi, i = 1, . . . , n. Find the expected number and the variance of the number of distinct types
Suppose that the joint probability mass function of X and Y is(a) Find the probability mass function of Y.(b) Find the probability mass function of X.(c) Find the probability mass function of Y −X.
Each member of a population is either type 1 with probability p1 or type 2 with probability p2 = 1 − p1. Independent of other pairs, two individuals of the same type will be friends with
If X is uniform over (0, 1), calculate E[Xn] and Var(Xn).
(a) Calculate E[X] for the maximum random variable of Exercise 37.(b) Calculate E[X] for X as in Exercise 33.(c) Calculate E[X] for X as in Exercise 34.
A coin, having probability p of landing heads, is flipped until a head appears for the rth time. Let N denote the number of flips required. Calculate E[N].Hint: There is an easy way of doing this. It
Let c be a constant. Show that(a) Var(cX) = c2Var(X);(b) Var(c + X) = Var(X).
Prove that E[X2] ≥ (E[X])2. When do we have equality?
For any event A, we define the random variable I {A}, called the indicator variable for A, by letting it equal 1 when A occurs and 0 when A does not. Now, if X(t) is a nonnegative random variable for
Consider three trials, each of which is either a success or not. Let X denote the number of successes. Suppose that E[X] = 1.8.(a) What is the largest possible value of P{X = 3}?(b) What is the
If X is a nonnegative integer valued random variable, show thatHint: Define the sequence of random variables In,n ≥ 1, byNow express X in terms of the In.(b) If X and Y are both nonnegative integer
A total of r keys are to be put, one at a time, in k boxes, with each key independently being put in box i with probability pi , k i=1 pi = 1. Each time a key is put in a nonempty box, we say that a
In Exercise 43, let Y denote the number of red balls chosen after the first but before the second black ball has been chosen.(a) Express Y as the sum of n random variables, each of which is equal to
An urn contains n+m balls, of which n are red and m are black. They are withdrawn from the urn, one at a time and without replacement. Let X be the number of red balls removed before the first black
Suppose that each coupon obtained is, independent of what has been previously obtained, equally likely to be any of m different types. Find the expected number of coupons one needs to obtain in order
Consider the case of arbitrary p in Exercise 29. Compute the expected number of changeovers.
Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability 1−p.The winner of the series is the first team
An urn has 8 red and 12 blue balls. Suppose that balls are chosen at random and removed from the urn, with the process stopping when all the red balls have been removed. Let X be the number of balls
Let X1, . . . , X10 be independent and identically distributed continuous random variables with distribution function F, and mean μ = E[Xi ]. Let X(1)
Let X1,X2, . . . , Xn be independent random variables, each having a uniform distribution over (0, 1). Let M = maximum (X1,X2, . . . , Xn). Show that the distribution function of M,FM(·), is given
A point is uniformly distributed within the disk of radius 1. That is, its density isFind the probability that its distance from the origin is less than x, 0 ≤ x ≤ 1. f(x, y) = C, 0 x + y1
The density of X is given byWhat is the distribution function of X? Find P{X>20}. f(x)= [10/x, |0, for x > 10 for x 10
Let the probability density of X be given by(a) What is the value of c?(b) P 1 2 =? [c(4x-2x), 0
Let X be a random variable with probability density(a) What is the value of c?(b) What is the cumulative distribution function of X? f(x)= |0, Jc(1-x), -1
Let X be a Poisson random variable with parameter λ. Show that P{X = i} increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer
Consider the following gambling game, which starts with you choosing one of the numbers 1, . . . , 6. Three fair dice are then rolled. If your number does not appear on any of the dice, you lose 1.
Consider n independent flips of a coin having probability p of landing heads.Say a changeover occurs whenever an outcome differs from the one preceding it. For instance, if the results of the flips
Suppose that we want to generate a random variable X that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some
A fair coin is independently flipped n times, k times by A and n−k times by B.Show that the probability that A and B flip the same number of heads is equal to the probability that there are a total
A total of n + 1 players are bidding for a 100 payoff, with the highest bid winning the amount by which 100 exceeds the bid. Player 1 knows that the bids of players 2 through n+1 are independent

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