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nonparametric statistical inference

Probability And Statistical Inference 3rd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

A fair coin is tossed 2n times. How large must n be if it is known that the probability of the equal number of heads and tails is less than 0.1?
Let X1, ...,X360 represent the outcomes of 360 tosses of a fair die. Let S360 be the total score X1 + ··· + X360, and for j = 1, ..., 6, let Yj be the total number of tosses that give outcome j. Use normal approximation to obtain: (i) P(55 < Y3 < 62). (ii) P(1,200 < S360 < 1,300). (iii)P(1,200 <
A regular dodecahedron (12-sided Platonian solid) has six red and six white faces, with the faces of each color labeled 1, . . . , 6. If you toss a face with label k you pay or win $k, depending on whether the color is red or white. Find the probability that after 50 tosses you are ahead by more
Let X1, ...,Xn be a random sample from the BETA(4, 2) distribution. Let Sn =X1 + ··· + Xn. Find the smallest n for which P{Sn ≥ 0.75n} ≤ 0.01.
Passengers on an international flight have a luggage weight limit B. The actual weight W of the passenger’s luggage is such that W/B has a BETA(a,b) distribution where a/(a + b)=0.9. Assume that the weights of luggage of different passengers are independent and that the plane has 220 seats. Find
Assume that 500 students at a certain college will graduate on a given day. Because of space limitations, the college offers each student two tickets for the commencement ceremony. From past experience, it is known that 50% of the students will invite two guests to attend the ceremony, 20% students
Let X1, ...,Xn be a random sample from the distribution with density f(x) = xe−x,x> 0. Find c if it is known that P{Xn > c} = 0.75 for n = 250.
Let X1, ...,Xn be a random sample from a logistic distribution with a cdf F(x) =1/(1 + e−x), and let Vn = Xn:n. Then Vn P→∞, but Vn − log n has a proper limiting distribution. Find limn→∞P{Vn − log n ≤ 0} and limn→∞P{|Vn − log n| ≤ 1}.
A random variable has a Pareto distribution with parameters a,b (a > 0,b> 0) if its density is f(x; a,b) = a b(1 + x/b)a+1 , x> 0.Let X1, ...,Xn be a random sample from the Pareto distribution with density f(x; 1, 1). (i) Find the limiting distribution of random variable Un = nX1:n.[Hint: Find cdf
Let X1, ...,Xn be a random sample from distribution with cdf F(x) = x2/9 for 0
Let X1, ...,Xn be a random sample from continuous distribution with a cdf F such that F −1 exists. Find (if it exists) the limiting distribution of: (i) Un = nF(X1:n).(ii) Wn = n[1 − F(Xn:n)]. (iii) Vn = nF(X3:n).
Let X1, ...,Xn be a random sample from distribution with cdf F(x)=1 − x−2 for 1 ≤ x < ∞, and 0 otherwise. Find the cdf of limiting distribution of: (i) X1:n. (ii) Xn 1:n.(iii) Xn:n/√n.
Let Z1, ...,Zn be a random sample from N(0, 1) distribution. Find the limiting distribution of Yn = n i=1(Zi + 1/n)/√n.
Let X1, ...,Xn be a random sample from a POI(λ) distribution. Show that e−Xn P→P(X1 = 0).
Genest (1987) provides the following algorithm for generating random samples from the so-called Frank family of bivariate distributions: (a) Generate two independent observations U1 and U2 from U[0, 1]. (b) Obtain T = αU1 + (α − αU1 )U2. (c) Let X = U1 and Y = logα[T/(T + (1 − α)U2)],
Kennedy and Gentle (1980) provide the following algorithm for generating a beta distribution: Generate U1 and U2—two independent observations from the U[0, 1]distribution. For α > 0 and β > 0 denote V1 = U α1 and V2 = Uβ2 . According to the Accept/Reject algorithm, let X = V1/(V1 + V2) if V1
The Box–Muller transformation of two independent, uniform variables into two independent standard normal variables was presented in Theorem 6.4.1. Another algorithm, proposed by Marsaglia and Bray (1964), is to generate U1 and U2 as two independent observations from U[−1, 1]. If V = U2 1 + U2 2
Apply an Accept/Reject method to the Laplace distribution with density f(x)=1.5e−3|x| to generate observations from a standard normal distribution. List the obtained values and specify how many of them you were able to obtain using the random sample of size 5 from U[0, 1]: 0.222795, 0.516174,
Generate a random sample from the Gompertz distribution with survival function S(t) = exp{1 − exp(2t)} using the following random sample from the U[0,1] distribution: 0.289365, 0.228349, 0.732889.
A generalized Laplace distribution has a density given by the formula f(x) = pλ1e−λ1x if x ≥ 0(1 − p)λ2eλ2x otherwise, (9.45)where λ1 > 0,λ2 > 0. Generate two independent observations from a generalized Laplace distribution with p = 1/4,λ1 = 3, and λ2 = 1/2, based on a random sample
The double exponential (or Laplace) distribution has a density given by the formula f(x)=(λ/2)e−λ|x| for −∞ 0. Obtain a random sample from the Laplace distribution with λ = 2 based on a random sample 0.744921 and 0.464001 from the U[0, 1] distribution.
Obtain a random sample of size 4 from a Pareto distribution with a density f(x) =(1 + x)−2 for x > 0 and 0 otherwise. Use the following random sample from U[0, 1]:0.187724, 0.386997, 0.182338, and 0.028113.
Obtain a sample of size 6 from a POI(2) distribution based on following six independent observations from U[0, 1] distribution: 0.090907, 0.185040, 0.124341, 0.299086, 0.428996, 0.927245.
Find π(r,n; Fθ) when Fθ is the U[0, θ] distribution. Find π(r,n; F) for the family F = {Fθ},θ> 0.
Let X1, ...,Xn and Y1, ...,Yn be two independent random samples from the same continuous distribution with a densityf. Show that P{Xi:n ≤ t} ≥ P{Yj:n ≤ t} for every t if and only if i ≤ j.
Let X1,X2 be a random sample from the U(0, 1) distribution. Find the probability that the values in the sample divide the (0, 1) interval into three parts, each of the length at least 0.3 (i.e., X1:2 > 0.3,X2:2 > X1:2 + 0.3, and X2:2 < 0.7).
Let X1, ...,X5 be a random sample from the BETA(2, 1) distribution. (i) Find the density of a joint distribution of X1:5,X2:5,X4:5. (ii) Obtain E(X2:5|X4:5).(iii) Obtain E(X4:5|X2:5). (iv) Find the distribution of Y = X2:5/X1:5. (v) Check if variables X1:n/Xn:n and Xn:n are independent.
Let X1, ...,Xn be a random sample from the U(0,θ) distribution. Find: (i) E(Xm 1:n)and E(Xm n:n). (ii) E(Xn:n|X1:n). (iii) The cdf of variable V = Xn:n − X1:n. (iv) The probability that the interval [X1:n,Xn:n] does not include a given point x0. (v) The point x0 for which the probability in part
Let X1, ...,Xn be a random sample from the U[0, 1] distribution. Find sample size n such that E(R)=0.75, where R = Xn:n − X1:n.
Use results from Example 9.3 to determine Var(Xl:n − Xk:n),l > k, in a random sample X1, ...,Xn from the U[0,θ] distribution.
Let X1,X2 be a random sample of size 2 from a continuous distribution with a median θ. (i) Find P(X1:2
Let X1, ...,Xn be a random sample selected from the U[0, 1] distribution. (i) Find the distribution of variable W = Xi:n − Xi−1:n, 1 < i ≤ n. (ii) Show that the distribution of Xj:n is BETA(j,n − j + 1) for j = 1, ...,n. (ii) Find the distribution of a sample median Xk+1:n, and obtain its
In a random sample X1, ...,Xn selected from EXP(λ) distribution find the density of: (i) X1:n. (ii) √nX1:n. (iii) X2:n.
Derive the formula for E(Xk), where X ∼ F(ν1,ν2). (Hint: Use similar approach as in Problem 9.2.6).
Derive formula (9.20). [Hint: Use formula (8.52) for the kth moment of a GAM(α,λ)distribution and the fact that variables X and U in (9.15) are independent.]
Show that variable aX/(1 + aX), where X ∼ F(ν1,ν2) and a = ν1/ν2, has a BETA(ν1/2,ν2/2) distribution.
Let X1 ∼ GAM(1,λ), X2 ∼ GAM(2,λ), X3 ∼ GAM(3,λ) be independent random variables. Find constant a such that variable Y = aX1/(X2 + X3) has an F distribution.
Let X,Y,W be independent random variables such that X ∼ N(0, 1), Y ∼ N(1, 1), and W ∼ N(2, 4), respectively. Find k such that
Show that if xα is the α-quantile of a random variable X with an Fν1,ν2 distribution, then 1/xα is the (1 − α)-quantile of a random variable with an Fν2,ν1 distribution.
Let X ∼ tν. Show that X2 has F1,ν distribution.
Let X1,X2 be a random sample from the BIN(1,p) distribution. Obtain the probability that the sample mean does not exceed the sample variance.
Let X1, X2, X3 be sample means in three independent samples of sizes n1,n2,n3, respectively. Each sample was obtained from the N(μ,σ2) distribution. Find the distribution of V1 = (1/3)(X1 + X2 + X3) and V2 = w1X1 + w2X2 + w3X3, where wi = ni/(n1 + n2 + n3).
Let X1, ...,Xn and Y1, ...,Ym be two random samples from distributions with means μ1 and μ2, respectively, and the same variance σ2. (i) Find E(X − Y ) and Var(X − Y ). (ii) Assuming that μ1 = μ2 and both samples are of equal size, find n = m such that P(|X − Y | > σ/4) ≤ 0.05.
Statistic Gk, defined for k = 1, 2 as Gk = 1 n(n − 1)n i=1 nj =1|Xi − Xj |k, was proposed as a measure of variation by Jordan (1869). Show that G2 = 2S2, where S2 is given by formula (9.1).
Let X1,X2, ... be iid with an exponential distribution. For any positive m and n find the distribution of the ratio
Let X have a symmetric beta distribution. Find α and β if the coefficient of variation(ratio of standard deviation and the mean) is k. Does a solution exist for any k?
Students are late to a particular class with probability that has BETA(2, 20) distribution. Find the probability that in a class of 25 students no more than 2 students will be late on a given day.
Let X have a distribution BETA(α,β). Find: (i) the distribution of Y = 1 − X. (ii)E{Xk(1 − X)m}.
Let variables X and Y be the weight and height of newborn children in the United States. The joint distribution of X and Y is bivariate normal with μX = 20,σX = 1.5 inches, μY = 7.7,σy = 2 pounds, and ρ = 0.8. Find: (i) The expected weight of a newborn child with a height 21 inches. (ii) The
Assume that variables X1 and X2 have a bivariate normal distribution with E(X1)=3,E(X2)=2, Var(X1)=4, Var(X2)=1 and ρ = −0.6.Find: (i) P{X1 ≤ 4|X2 = 3}. (ii) P{|X2 − 1| ≥ 1.5|X1 = 2}.
Assume that variables X1,X2,X3 are independent and each has N(2, 1) distribution.Moreover let Y1,Y2,Y2, also be independent and each have N(2, 3) distribution. Find P(3X1 + 2X2 + X3 > Y1 + 2Y2 + Y3).
Assume that X1 and X2 are independent, with N(3, 6) and N(−1, 2) distributions, respectively. Find: (i) P(3X1 − 2X2 ≥14). (ii) P(X1 < X2).
Find γ3—the skewness of a lognormal distribution with parameters μ and σ2.
A “100-year water,” or flood, is the water level that is exceeded once in a 100 years(on average). Suppose that the threatening water levels occur once a year and have a normal distribution. Suppose also that at some location the 100-year water means the level of 30 feet above average. What is
Let random variable X have a N(μ,σ2) distribution. Find: (i) μ if σ2 = 2 and P(X ≤12) = 0.72. (ii) σ2 if μ = 2 and P(X ≥5) = 0.39.
Find P(|X − 2| ≤ 0.5) if X ∼ N(1, 4).
Determine x in the following cases: (i) Φ(x) = 0.62. (ii) Φ(x) = 0.45. (iii) P(|Z| ≤ x)= 0.98. (iv) P(1.4 ≤ Z ≤ x)=0.12.
Use the tables of normal distribution or any statistical software to determine the probabilities: (i) P(0 ≤ Z ≤ 1.34). (ii) P(0.14 ≤ Z ≤ 2.01). (iii)P(−0.21 ≤ Z ≤ −0.04). (iv) P(−0.87 ≤ Z ≤ 1.14). (v) P(|Z| ≥ 1.02). (vi)P(Z ≥ 1.11).
It was found that the survival time (in years) in a group of patients who went through a certain medical treatment and are in the similar risk group follows WEI(2, 1/3) distribution. Find: (i) The median survival time for such patients. (ii) The probability that a randomly selected patient will
(i) Find the median and the mode of the Weibull distribution with a density (8.57). (ii)Prove the formula (8.58).
Find the distribution of X = θ(− log U)1/k, if U ∼ U(0, 1).
For a variable X having Laplace distribution with density f(x, 0.4, 1, 2) given by (8.55), find: (i) Mean; (ii) Variance; (iii) Median; (iv) Interquartile range.
For a variable X that has a Laplace distribution with λ = 1, find: (i)Interquartile range(the difference between the upper and lower quartile). (ii) Variance. (iii) Kurtosis.
Suppose that patients arrive to the emergency room according to Poisson distribution with mean λ = 3. What is the expected time until the 9th patient arrives? Find the probability that the next (10th) patient arrives 1 hour later.
In the flowchart of Figure 8.5, the block denoted “sample U” means that the computer samples a value of random variable U with a distribution uniform on (0, 1), the samplings being independent each time the program executes this instruction.Assume that m is a positive integer and λ > 0. Find:
A system consists of five components. Suppose that the lifetimes of the components are independent, with exponential distributions EXP(λ1), . . . , EXP(λ5). Find the cdf and density of variable T = time to failure of the system if the components are connected:(i) In series (see Figure 8.2), so
Show that if variable X has a GAM(n, 1) distribution, where n is a positive integer, then its cdf is given by the following formula:FX(x) = 1Γ(n)x 0t n−1e−t dt = 1 − e−x n−1 j=0 xj j!.(Hint: Integrate by parts and use induction.)
Consider two independent Poisson processes with the same parameter λ. Let Ni(t) ,i = 1, 2 be the number of events in ith process which occurred up to time t, and let UT be the set of all those times t with 0≤ t ≤ T at which N1(t) = N2(t).Find E(UT ) given that: (i) N1(T) = N2(T)=2. (ii)
Assume that chocolate chips are distributed within a cake according to a Poisson process with parameter λ. A cake is divided into two parts of equal volume (disregard the possibility of cutting through a chocolate chip). Show that the probability that each part of the cake has the same number of
Two parts of a document are typed by two typists. Let X and Y be the numbers of typing errors in the two parts of the paper. Assuming that X and Y are independent and have Poisson distributions with parameters λ1 and λ2, respectively, find the probability that: (i) The paper (i.e., two combined
Traffic accidents at a given intersection occur following a Poisson process. (i) Given that 10 accidents occurred in June, what is the probability that the seventh accident occurred before June 10? (ii) If it is known that n accidents occurred in April, what is the expected number of accidents that
Suppose that the number of eggs X laid by a bird has a Poisson distribution. Each egg hatches with probability p, independently of what happens to other eggs. Let V1 and V2,V1 + V2 = X, denote the numbers of eggs that hatch, and the number of eggs that do not hatch, respectively. Show that V1 and
Let X be the number of failures preceding the rth success in a sequence of Bernoulli trials with probability of success p. Show that if q → 0,r → ∞ in such a way that rq = λ > 0, then P{X = k} → λk k!e−λfor every k = 0, 1, 2, . . . . This shows that the negative binomial distribution
(ii), let X be an integer-valued random variable such that X = X1 + X2, where X1,X2 are independent, identically distributed integer-valued random variables. Show that P{X is even } ≥ 0.5 (this property has been pointed out to us by Steve MacEachern, personal communication).
(Does Nature Prefer Even Numbers?) Generalizing Problem
Find the approximate probability that in 1,000 randomly chosen persons there are exactly: (i) Two born on New Year and two born on Christmas. (ii) Four born on either Christmas or New Year.
Weekly numbers of traffic accidents at intersections A, B, and C are independent, each with a Poisson distribution. It is known that, on the average, the number of accidents at intersection A is the same as the number of accidents at intersections B and C combined, while the average number of
A certain store makes, on average, two sales per hour between 9:00 a.m. and 2:00 p.m., and three sales per hour between 2:00 p.m. and 9:00 p.m. The numbers of sales in different time periods are independent and have a Poisson distribution. Find: (i) the probability of more than three sales between
Suppose that the daily numbers of ships arriving to a certain port are independent, each with POI(3) distribution. Find: (i) the expected number of days in April when there are no arrivals; (ii) the expected number and variance of days during the summer months (June, July, August) with the number
Accidents in a given plant occur at a rate of 1.5 per month. The numbers of accidents in different months are independent and follow the Poisson distribution. Find the probability of: (i) five accidents in a period of five consecutive months; (ii) one accident in each of five consecutive months.
A book with 500 pages contains, on average, 3 misprints per 10 pages. What is the probability that there will be more than one page containing at least three misprints?
Let X have the POI(λ) distribution. Find: (i) the mode of X (i.e., the most likely value of X); (ii) P(Xis even). (Hint: Write the Taylor expansions for eλ and e−λ.Any ideas?)
Instead of (8.23), write X = η1 + η2 + ··· + ηa, where ηj = 1 or 0 depending on whether or not the jth element representing success was selected. Use this representation to derive formulas for the mean and variance of X as given in Theorem 8.2.2.
if instead of three winning tickets in each package, 30% of all tickets are winning and 10 tickets for each package are selected at random.
Answer (i)–(iii) in Problem
Kim and Jason bought a package of 10 lottery tickets, knowing that each package had to include three winning tickets. Since Kim paid 60% of the total cost, while Jason paid 40%, they took six and four tickets, respectively. Find the probability that: (i) Kim got at least one winning ticket; (ii)
An urn contains six chips, three red and three green. Four chips are selected without replacement. Find E(X) and Var(X), where X = number of red chips in the sample.
An urn contains nine chips, five of them red and four blue. Three chips are drawn without replacement. Find the distribution of X = number of red chips drawn.
Show that if X has a negative binomial distribution NBIN(r,p), then E[(r − 1)/(r + X − 1)] = p.
In the flowchart in Figure 8.1, m > 0 is an integer and 0
Assume that in a tennis match between A and B, the probability of winning a tennis set by player A is p and that the results of sets are independent. Let T be the number of sets played in a match. Find the distribution of T and E(T) as a function of p, assuming that the match is played by: (i) men,
A hospital needs 20 volunteers for the control group in testing the efficiency of some treatment. The candidates are subject to psychological and medical screening, and on average, only 1 in 15 candidates is found acceptable for the experiment. The cost of screening, whether or not a candidate is
Assume that the probability that a birth is a multiple one (twins, triplets, etc.) isπ. Given that a birth is a multiple one, probabilities α2,α3, ... of twins, triplets,. . . satisfy the condition αk+1 = γαk. Find π,α2, and γ if it is known that the expected number of children born in 100
Assume that the probability of twins being identical is β, and that the sexes of children are determined independently, with probability of a boy being b (possibly b = 1/2).Find the expected number of twin births recorded in the hospital before the first pair of: (i) boys, (ii) girls, and (iii)
Six dice are tossed simultaneously until, for the first time, all of them show the same face. Find E(U) and Var(U), where U is the number of tosses until this happens.
Show that for the binomial distribution we have P{Sn ≤ k} = (n − k)n k1−p 0xn−k−1(1 − x)k dx.
Show that if Sn is a binomial random variable, then for k = 1, 2, ...,n, P{Sn = k} = (n − k + 1)p k(1 − p) P{Sn = k − 1}.
Two players (or two teams) are negotiating the rules for determining the championship. The two possibilities are “best of five” or “best of seven.” This means that whoever wins three (respectively, four) games is the champion. Assume that games are independent and that the probability of
A regular die is tossed until the same side turns up twice in a row. Find: (i) the probability that it will happen in tosses 8 and 9. (ii) The probability that it will happen in tosses n and n + 1.
Two balanced coins are tossed together independently. Find: (i) the distribution of the number of tosses required for both coins to show the same side; and (ii) the distribution of the number of tosses needed to get at least one head.

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