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statistical techniques in business
Questions and Answers of
Statistical Techniques in Business
Let X be a continuous random variable with cdf F(x). Suppose Y = X+Δ, where Δ > 0. Show that Y is stochastically larger than X.
Suppose X is a random variable with mean 0 and variance σ2. Recall that the function Fx,ϵ(t) is the cdf of the random variable U = I1−ϵX + [1 − I1−ϵ]W, where X, 1−ϵ, and W are
Suppose that the hypothesis H0 concerns the independence of two random variables X and Y . That is, we wish to test H0 : F(x, y) = F1(x)F2(y), where F, F1, and F2 are the respective joint and
Let the scores a(i) be generated by aϕ(i) = ϕ[i/(n+ 1)], for i = 1, . . . , n, where ∫10 ϕ(u) du = 0 and ∫10 ϕ2(u) du = 1. Using Riemann sums, with subintervals of equal length, of the
Suppose the random variable e has cdf F(t). Let ϕ(u) =√12[u − (1/2)],0 < u < 1, denote the Wilcoxon score function.(a) Show that the random variable ϕ[F(ei)] has mean 0 and variance 1.(b)
The following amounts are bets on horses A,B,C,D, and E to win.Suppose the track wants to take 20% off the top, namely, $200,000. Determine the payoff for winning with a $2 bet on each of the five
Let Y have a binomial distribution in which n = 20 and p = θ. The prior probabilities on θ are P(θ = 0.3) = 2/3 and P(θ = 0.5) = 1/3. If y = 9, what are the posterior probabilities for θ = 0.3
Consider the Bayes modelBy performing the following steps, obtain the empirical Bayes estimate of θ.(a) Obtain the likelihood function(b) Obtain the mle ^β of β for the likelihood m(x|β).(c) Show
Show that P(C) = 1.
Show that P(Cc) = 1 − P(C).
Let X1,X2, . . . , Xn denote a random sample from a Poisson distribution with mean θ, 0 < θ < ∞. Let Y = Σn1 Xi. Use the loss function L[θ, δ(y)] = [θ−δ(y)]2. Let θ be an observed
Show that if C1 ⊂ C2 and C2 ⊂ C1 (that is, C1 ≡ C2), then P(C1) = P(C2).
Show that if C1, C2, and C3 are mutually exclusive, then P(C1∪C2∪C3) = P(C1) + P(C2) + P(C3).
Show that P(C1 ∪ C2) = P(C1) + P(C2) − P(C1 ∩ C2).
Consider the following mixed discrete-continuous pdf for a random vector (X, Y), (discussed in Casella and George, 1992):for α > 0 and β > 0.(a) Show that this function is indeed a joint,
If computation facilities are available, write a program for the Gibbs sampler of Exercise 11.4.7. Run your program for α = 10, β = 4, m = 3000, and n = 6000. Obtain estimates (and confidence
Let Y4 be the largest order statistic of a sample of size n = 4 from a distribution with uniform pdf f(x; θ) = 1/θ, 0 < x < θ, zero elsewhere. If the prior pdf of the parameter g(θ) =
Calculate the lim and ¯lim of each of the following sequences:(a) For n = 1, 2, . . ., an = (−1)n (2 − 4/2n).(b) For n = 1, 2, . . ., an = ncos(πn/2).(c) For n = 1, 2, . . ., an = 1/n + cos
Let {an} and {dn} be sequences of real numbers. Show that lim (an + dn) lim an + lim dn. n→∞ n→∞ n→∞
Let {an} be a sequence of real numbers. Suppose {ank} is a subsequence of {an}. If {ank} → a0 as k→∞, show that limn→∞ an ≤ a0 ≤ ¯limn→∞ an.
For the test at level 0.05 of the hypotheses given by (4.6.1) with μ0 = 30,000 and n = 20, obtain the power function, (use σ = 5000). Evaluate the power function for the following values: μ =
Define the sets A1 = {x : −∞ < x ≤ 0}, Ai = {x : i − 2 < x ≤ i − 1}, i = 2, . . . , 7, and A8 = {x : 6 < x < ∞}. A certain hypothesis assigns probabilities pi0 to these sets
A die was cast n = 120 independent times and the following data resulted:If we use a chi-square test, for what values of b would the hypothesis that the die is unbiased be rejected at the 0.025
A number is to be selected from the interval {x : 0 < x < 2} by a random process. Let Ai = {x : (i − 1)/2 < x ≤ i/2}, i = 1, 2, 3, and let A4 = {x :3/2 < x < 2}. For i = 1, 2, 3,
Consider the sample of data:(a) Obtain the five-number summary of these data.(b) Determine if there are any outliers.(c) Boxplot the data. Comment on the plot. 13 5 202 15 99 4 67 83 36 11 301 23 213
Let Y2 and Yn−1 denote the second and the (n − 1)st order statistics of a random sample of size n from a distribution of the continuous type having a distribution function F(x). Compute
Suppose X is a random variable with the pdf fX(x) = b−1f((x − a)/b), where b > 0. Suppose we can generate observations from f(z). Explain how we can generate observations from fX(x).
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from a distribution of the continuous type having distribution function F(x).(a) What is the distribution of
Two different teaching procedures were used on two different groups of students. Each group contained 100 students of about the same ability. At the end of the term, an evaluating team assigned a
Consider the problem from genetics of crossing two types of peas. The Mendelian theory states that the probabilities of the classifications (a) Round and yellow, (b) Wrinkled and
Let Y1 < Y2 < · · · < Y10 be the order statistics of a random sample from a continuous-type distribution with distribution function F(x). What is the joint distribution of V1 = F(Y4) −
Assume that the weight of cereal in a “10-ounce box” is N(μ, σ2). To test H0 : μ = 10.1 against H1 : μ > 10.1, we take a random sample of size n = 16 and observe that ¯x = 10.4 and s =
Determine a method to generate random observations for the following pdf:If access is available, write an R function which returns a random sample of observations from this pdf. f(x) = { 42³ 0 0
Let f(x) = 1/6, x = 1, 2, 3, 4, 5, 6, zero elsewhere, be the pmf of a distribution of the discrete type. Show that the pmf of the smallest observation of a random sample of size 5 from this
Let the result of a random experiment be classified as one of the mutually exclusive and exhaustive ways A1,A2,A3 and also as one of the mutually exclusive and exhaustive ways B1,B2,B3,B4. Two
Let the result of a random experiment be classified as one of the mutually exclusive and exhaustive ways A1,A2,A3 and also as one of the mutually exhaustive ways B1,B2,B3,B4. Say that 180 independent
Each of 51 golfers hit three golf balls of brand X and three golf balls of brand Y in a random order. Let Xi and Yi equal the averages of the distances traveled by the brand X and brand Y golf balls
It is proposed to fit the Poisson distribution to the following data:(a) Compute the corresponding chi-square goodness-of-fit statistic.(b) How many degrees of freedom are associated with this
A certain genetic model suggests that the probabilities of a particular trinomial distribution are, respectively, p1 = p2, p2 = 2p(1−p), and p3 = (1−p)2, where 0 < p < 1. If X1,X2,X3
Let us say the life of a tire in miles, say X, is normally distributed with mean θ and standard deviation 5000. Past experience indicates that θ = 30,000. The manufacturer claims that the tires
Suppose we are interested in a particular Weibull distribution with pdfDetermine a method to generate random observations from this Weibull distribution. If access is available, write an R function
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere. Show that the kth order statistic Yk
Let z∗ be drawn at random from the discrete distribution which has mass n−1 at each point zi = xi − ¯x + μ0, where (x1, x2, . . . , xn) is the realization of a random sample. Determine
Let Y be b(300, p). If the observed value of Y is y = 75, find an approximate 90% confidence interval for p.
Suppose a random sample of size 2 is obtained from a distribution that has pdf f(x) = 2(1 − x), 0 < x < 1, zero elsewhere. Compute the probability that one sample observation is at least
Let Y1 < Y2 be the order statistics of a random sample of size 2 from a distribution of the continuous type which has pdf f(x) such that f(x) > 0, provided that x ≥ 0, and f(x) = 0 elsewhere.
It is known that a random variable X has a Poisson distribution with parameter μ. A sample of 200 observations from this distribution has a mean equal to 3.4. Construct an approximate 90% confidence
Let X1,X2, . . .,Xn be a random sample from N(μ, σ2), where both parameters μ and σ2 are unknown. A confidence interval for σ2 can be found as follows. We know that (n − 1)S2/σ2 is a random
Two numbers are selected at random from the interval (0, 1). If these values are uniformly and independently distributed, by cutting the interval at these numbers, compute the probability that the
Let X1,X2, . . . , Xn be a random sample from a gamma distribution with known parameter α = 3 and unknown β > 0. Discuss the construction of a confidence interval for β.
When 100 tacks were thrown on a table, 60 of them landed point up. Obtain a 95% confidence interval for the probability that a tack of this type lands point up. Assume independence.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from the exponential distribution with pdf f(x) = e−x, 0 < x < ∞, zero elsewhere.(a) Show that Z1 =
Let two independent random variables, Y1 and Y2, with binomial distributions that have parameters n1 = n2 = 100, p1, and p2, respectively, be observed to be equal to y1 = 50 and y2 = 40. Determine an
In the Program Evaluation and Review Technique (PERT), we are interested in the total time to complete a project that is comprised of a large number of subprojects. For illustration, let X1, X2, X3
Let Y1 < Y2 < Y3 < Y4 < Y5 denote the order statistics of a random sample of size 5 from a distribution of the continuous type. Compute:(a) P(Y1 < ξ0.5 < Y5).(b) P(Y1 < ξ0.25
Compute P(Y3 < ξ0.5 < Y7) if Y1 < · · · < Y9 are the order statistics of a random sample of size 9 from a distribution of the continuous type.
Let {Xn} be a sequence of p-dimensional random vectors. Show thatfor all vectors a ∈ Rp. D XnNp(μ, Σ) if and only if a'XnN₁(a'µ, a'Σa),
Let y1 < y2 < y3 be the observed values of the order statistics of a random sample of size n = 3 from a continuous type distribution. Without knowing these values, a statistician is given these
Let Y1 < Y2 < · · · < Yn denote the order statistics of a random sample of size n from a distribution that has pdf f(x) = 3x2 θ3, 0 < x < θ, zero elsewhere.(a) Show that P(c <
Let ‾X denote the mean of a random sample of size 100 from a distribution that is χ2(50). Compute an approximate value of P(49 < ‾X < 51).
Let Xn and Yn be p-dimensional random vectors. Show that ifwhere X is a p-dimensional random vector, then Yn D→ X. P Xn - Yn 0 and XnDX,
Let {an} be a sequence of real numbers. Hence, we can also say that {an} is a sequence of constant (degenerate) random variables. Let a be a real number. Show that an → a is equivalent to an P→ a.
Let ‾X denote the mean of a random sample of size 128 from a gamma distribution with α = 2 and β = 4. Approximate P(7 < ‾X < 9).
Let X1, . . . , Xn be a random sample from a uniform(a, b) distribution. Let Y1 = min Xi and let Y2 = max Xi. Show that (Y1, Y2)' converges in probability to the vector (a, b)'.
Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) = e−(x−θ), θ < x < ∞, zero elsewhere. Let Zn = n(Y1 − θ). Investigate the limiting
Let Y denote the sum of the observations of a random sample of size 12 from a distribution having pmf p(x) = 1/6, x = 1, 2, 3, 4, 5, 6, zero elsewhere. Compute an approximate value of P(36 ≤ Y ≤
Suppose Xn has a Np(μn,Σn) distribution. Show that Χn 5 Ν,(μ, Σ) if μη → μ and Ση – Σ.
Let Xn and Yn be p-dimensional random vectors such that Xn and Yn are independent for each n and their mgfs exist. Show that ifwhere X and Y are p-dimensional random vectors, then (Xn, Yn) D→
Compute an approximate probability that the mean of a random sample of size 15 from a distribution having pdf f(x) = 3x2, 0 < x < 1, zero elsewhere, is between 3/5 and 4/5.
Let the pmf of Yn be pn(y) = 1, y = n, zero elsewhere. Show that Yn does not have a limiting distribution. (In this case, the probability has “escaped” to infinity.)
Let Y be b(400, 1/5). Compute an approximate value of P(0.25 < Y/400).
Let X1, X2, . . . , Xn be a random sample of size n from a distribution that is N(μ, σ2), where σ2 > 0. Show that the sum Zn =Σn1 Xi does not have a limiting distribution.
If Y is b(100, 1/2), approximate the value of P(Y = 50).
Let f(x) = 1/x2, 1 < x < ∞, zero elsewhere, be the pdf of a random variable X. Consider a random sample of size 72 from the distribution having this pdf. Compute approximately the probability
Forty-eight measurements are recorded to several decimal places. Each of these 48 numbers is rounded off to the nearest integer. The sum of the original 48 numbers is approximated by the sum of these
Let the random variable Zn have a Poisson distribution with parameter μ = n. Show that the limiting distribution of the random variable Yn = (Zn−n)/√n is normal with mean zero and variance 1.
Let X1,X2, . . .,Xn be a random sample from a Poisson distribution with mean μ. Thus, Y = Σni=1 Xi has a Poisson distribution with mean nμ. Moreover, ‾X = Y/n is approximately N(μ, μ/n) for
Let X1,X2, . . . , Xn be a random sample from a Γ(α = 3, β = θ) distribution, 0 < θ < ∞. Determine the mle of θ.
Let X1,X2, . . .,Xn represent a random sample from each of the distributions having the following pdfs:(a) f(x; θ) = θxθ−1, 0 < x < 1, 0 < θ < ∞, zero elsewhere.(b) f(x; θ) =
Given f(x; θ) = 1/θ, 0 < x < θ, zero elsewhere, with θ > 0, formally compute the reciprocal ofCompare this with the variance of (n+1)Yn/n, where Yn is the largest observation of a random
Let X1,X2, . . . , Xn be a random sample from the distribution N(θ1, θ2). Show that the likelihood ratio principle for testing H0 : θ2 = θ'2 specified, and θ1 unspecified against H1 : θ2 ≠
Let X1,X2, . . . , Xn be a random sample from a N(μ0, σ2 = θ) distribution, where 0 < θ < ∞and μ0 is known. Show that the likelihood ratio test of H0 : θ = θ0 versus H1 : θ ≠ θ0
Suppose X1,X2, . . . , Xn1 is a random sample from a N(θ, 1) distribution. Besides these n1 observable items, suppose there are n2 missing items, which we denote by Z1, Z2, . . ., Zn2 . Show that
For the test described in Exercise 6.3.5, obtain the distribution of the test statistic under general alternatives. If computational facilities are available, sketch this power curve for the case
Consider two Bernoulli distributions with unknown parameters p1 and p2. If Y and Z equal the numbers of successes in two independent random samples, each of size n, from the respective distributions,
The following data are observations of the random variable X = (1−W)Y1+ WY2, where W has a Bernoulli distribution with probability of success 0.70; Y1 has a N(100, 202) distribution; Y2 has a
If X1,X2, . . . , Xn is a random sample from a distribution with pdfshow that Y = 2 ‾X is an unbiased estimator of θ and determine its efficiency. f(x; 0) = { 303 (x+0)4 0 0
Let X1,X2, . . . , Xn be a random sample from a N(θ, σ2) distribution, where σ2 is fixed but −∞ < θ < ∞.(a) Show that the mle of θ is ‾X.(b) If θ is restricted by 0 ≤ θ <
Let X1,X2, . . . , Xn be a random sample from a Γ(α = 3, β = θ) distribution, where 0 < θ < ∞.(a) Show that the likelihood ratio test of H0 : θ = θ0 versus H1 : θ ≠ θ0 is based
Let X1,X2, . . .,Xn be a random sample from the Poisson distribution with 0 < θ ≤ 2. Show that the mle of θ is ^θ = min{‾X, 2}.
Let X1,X2, . . .,Xn be a random sample from a distribution with pdf f(x; θ) = θ exp{−|x|θ} /2Γ(1/θ), −∞ < x < ∞, where θ > 0. Suppose Ω = {θ : θ = 1, 2}. Consider the
Let X1,X2, . . .,Xn be a random sample from the beta distribution with α = β = θ and Ω = {θ : θ = 1, 2}. Show that the likelihood ratio test statistic Λ for testing H0 : θ = 1 versus H1 : θ
Let S2 be the sample variance of a random sample of size n > 1 from N(μ, θ), 0 < θ < ∞, where μ is known. We know E(S2) = θ.(a) What is the efficiency of S2?(b) Under these
Let X1,X2, . . .,Xn be a random sample from a Poisson distribution with mean θ > 0. Test H0 : θ = 2 against H1 : θ ≠ 2 using(a) −2 logΛ.(b) A Wald-type statistic.(c) Rao’s score
Let X1,X2, . . .,Xn be a random sample from a Γ(α, β) distribution where α is known and β > 0. Determine the likelihood ratio test for H0 : β = β0 against H1 : β ≠ β0.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample from a uniform distribution on (0, θ), where θ > 0.(a) Show that Λ for testing H0 : θ = θ0 against H1 : θ ≠
Let the number X of accidents have a Poisson distribution with mean λθ. Suppose λ, the liability to have an accident, has, given θ, a gamma pdf with parameters α = h and β = h−1; and θ, an
If the variance of the random variable X exists, show that E(X²) > [E(X)]².
From a well-shuffled deck of ordinary playing cards, four cards are turned over one at a time without replacement. What is the probability that the spades and red cards alternate?
Let pX(x) be the pmf of a random variable X. Find the cdf F(x) of X and sketch its graph along with that of pX(x) if:(a) pX(x) = 1, x = 0, zero elsewhere.(b) pX(x) = 1/3, x = −1, 0, 1, zero
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