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statistics principles and methods
Mathematical Statistics For Economics And Business 2nd Edition Ron C.Mittelhammer - Solutions
A Seattle newspaper intends to administer two different surveys relating to two different anti-tax initiatives on the ballot in November. The proportion of surveys mailed that will actually be completed and returned to the newspaper can be represented as the outcome of a bivariate random variable
An automobile dealership sells two types of four-door sedans, the "Land Yacht" and the "Mini-Rover." The number of Land Yachts and Mini-Rovers sold on any given day varies, with the probabilities of the various possible sales outcomes given by the following table: Land Yachts sell for \$22,000
The season average price per pound, \(p\), and total season quantity sold, \(q\), of sweet cherries in a regional market can be represented as the outcome of a bivariate random variable \((P, Q)\) with the joint probability density function \(f(p, q)=.5 q e^{-q(.5+p)} I_{(0, \infty)}(q) I_{(0,
The yield/acre of wheat on a given parcel of land can be represented as the outcome of a random variable \(Y\) defined by \(Y=10 x^{1 / 3} e^{\varepsilon}\) for \(x \in[8,100]\), where\(Y=\) wheat output in bushes/acre\(x=\) pounds/acre of fertilizer applied\(\varepsilon=\) is a random variable
Let \(X\) have the moment generating function \(M_{X}(t)\). Show that(a) \(M_{(X+a)}(t)=e^{a t} M_{X}(t)\).(b) \(M_{b X}(t)=M_{X}(b t)\).(c) \(M_{(X+a) / b}(t)=e^{(a /b) t} M_{X}(t / b)\).
The AJAX Disk Co. manufactures compact disks (CDs) for the music industry. As part of its quality-control program, the diameter of each disk is measured using an electronic measuring device. Letting \(X_{1}\) represent the actual diameter of the disk and \(X_{2}\) represent the measured diameter of
An investor wishes to invest \(\$ 1,000\) and is examining two investment prospects. The net dollar return per dollar invested in the two projects can be represented as the outcome of a bivariate random variable \(\left(X_{1}, X_{2}ight)\) where\(\mathrm{E}\left[\begin{array}{l}X_{1} \\
The length of time in minutes for an individual to be served at a local restaurant is the outcome of a random variable, \(T\), having a mean of 6 and a variance of 1.5 . How probable is the event that an individual will be served within 3 to \(9 \mathrm{~min}\) ?
(a) Find the moment-generating function of a random variable \(X\) having the density function\(f(x)=\frac{1}{8}\left(\begin{array}{c}3 \\ x\end{array}ight) I_{(0,1,2,3)}(x)\).(Hint: Use of the binomial theorem may be helpful in finding a compact representation of this function.)Use the MGF to
The Rockbed Insurance Company sells 1-year term life insurance policies for \(\$ 10,000\) of annual coverage, where a 1 -year premium is charged to put the policy in force, and then if the insured person does not live through the year, his or her estate is paid \(\$ 10,000\). The mortality tables
The daily wholesale price and quantity sold of ethanol in a Midwestern regional market during the summer months is represented by the outcome of a bivariate random variable \((P, Q)\) having the following probability model \(\{R(P, Q), f(p, q)\}\) :\(f(p, q)= \begin{cases}.5 p e^{-p q} \text { for
Define the mean, median, mode, and .10 and .90 quantiles of the random variable \(X\) defined in the probability models \(\{R(X), f(x)\}\) below: (a) f(x) = x + 4 50 (b) f(x) = .5e */1(R(x)} (x), R(X) = [0, ) (c) f(x) = 3xI(R(x)} (x), R(X) = [0, 1] (d) f(x) = .05 (95)*-1{R(x)} (x), R(X) =
The regression curve of daily quantity demanded of tablet computers in a Midwestern market, measured in thousands of units and expressed as a function of price, all else held equal, is given by\(\mathrm{E}(\mathrm{Q} \mid p)=10 p^{-.5}\)The marginal probability density function of price is given
The yield per acre of a certain dwarf watermelon is highly dependent on the amount of rainfall that occurs during the growing season. Following standard cultivation practices, the relationship between tons per acre, \(Y\), and inches of rainfall, \(R\), is given by\(Y=25+2 R-.05 R^{2}\) for \(R
For each probability density function below, determine the mean and variance, if they exist, and define the median and mode. (a) f(x) = [(1+x)](-00,00)(x) (b) f(x) = 4x [0,1](x) (c) f(x) = .3(.7) 11.2.3...) (x) X (d) f(x) = (1,210) (x) 55
The daily dollar sales of a large retail "Big Box" store, measured in 1,000 dollar units, is a random variable, \(D\), that has an expectation of 20.(a) Provide an upper bound to the probability that dollar sales exceed 40,000 dollars on a given day.(b) If the variance of \(\mathrm{D}\) is 4,
Given the following three moments about the origin, derive the first three moments about the mean, and determine whether the random variable has a probability density function that is skewed, if you can. (a) E(X)=.2, E(X2) = .2, E(X) = .2 (b) E(X) = 1, E(X2) = 2, E(X) = 5 (c) E(X) = 2,
The bivariate random variable (P,Q) represents the weekly price, in dollars, and the quantity, in number of kegs, of an India Pale Ale beer, HopMeister, sold in the Pacific Northwest market. The moment generating function associated with this bivariate random variable is given by (a) What is the
Derive the moment generating function of each of the random variables below, and use it to define the mean and variance of the random variable.(a) \(f(x)=.2 e^{-.2 x} I_{(0, \infty)}(x)\)(b) \(f(x)=2 x I_{(0,1)}(x)\)(c) \(f(x)=.3^{x} \cdot 7^{1-x} I_{\{0,1\}}(x)\)
A small manufacturing firm produces and sells a product in a market where output prices are uncertain. The owner of the firm wishes to make a short run production decision that will maximize her expected utility, defined by\(\mathrm{E}(U(\pi))=\mathrm{E}(\pi)-\alpha[\operatorname{var}(\pi)]\)where
The weekly number of MACs and PCs sold by a salesperson at the local computer store can be represented as the outcome of a bivariate random variable \((X, Y)\) with the nonzero values of its joint probability density function given by the following table of probabilities:( \(\mathrm{X}\) is the
An investor has \(\$ 10,000\) to invest between two investment projects. The rate of return per dollar invested in the two projects can be represented as the outcome of a bivariate random variable \(\left(X_{1}, X_{2}ight)\) where\(\mathrm{E}\left[\begin{array}{l}X_{1} \\
The mean vector and covariance matrix of the trivariate random variable \(\mathbf{X}\) is given by\(\mathrm{E}(\mathbf{X})=\left[\begin{array}{c}-2 \\ 4 \\ 2\end{array}ight]\) and \(\boldsymbol{\operatorname { C o v }}(\mathbf{X})=\left[\begin{array}{ccc}10 & 2 & 1 \\ 2 & 5 & 0 \\ 1 & 0 &
The bivariate random variable (Y,X)(Y,X) has the following mean vector and covariance matrix: E[XY]=[105]E[XY]=[105] and Cov(X,Y)=[5222]Cov(X,Y)=[5222] (a) Derive the values of aa and bb in ^Y=a+bXY^=a+bX that minimize the expected squared distance between YY and ^YY^, i.e., that produce the
A shipment of 100 DVDs contains k defective disks. You randomly sample 20 DVDs, without replacement, from the shipment of 100 DVDs. Letting p = k/100, the probability that you will obtain less than three defective disks in your sample of 20 disks is then given by 3 P(x3)= x=0 True or False? (20) px
The daily price, \(p\), and quantity, demanded, \(q\), of gasoline on a European Wholesale spot market can be viewed (approximately) as the outcome of a bivariate normal random variable, where the bivariate normal density has mean vector and covariance matrix as
Show that the following probability density functions are members of the exponential class of densities:(a) Binomial family, for a fixed value of \(n\).(b) Poisson family.(c) Negative binomial family for a fixed value of \(r\).(d) Multinomial family, for a fixed value of \(n\).(e) Beta family.
For each PDF family below, show whether or not the family belongs to the exponential class of densities.(a) \(f(x ; \beta)=\beta x^{-(\beta+1)} I_{(1, \infty)}(x), \beta \in \Omega=(0, \infty)\). (This is a subfamily of the Pareto family of PDFs.)(b) \(f(x ; \Theta)=(1 /(2 \Theta)) \exp (-|x| /
Prove that if \(X\) has the geometric density, then the "memoryless property" \(P(x>s+t \quad \mid x>s)=P(x>t)\) holds for every choice of positive integers \(s\) and \(t\).
Prove that the CDFCDF of the geometric family of densities can be defined by F(b)=[1−(1−p)trunc(b)]F(b)=[1−(1−p)trunc(b)] I[1,∞)(b)I[1,∞)(b).
The quantity of wheat demanded, per day, in a midwestern market during a certain marketing period is represented by\(Q=100,000-12,500 P+V\) for \(\mathrm{p} \in[2,6]\), where\(Q\) is quantity demanded in bushels; \(p\) is price/bushel; and\(V\) is approximately normally distributed.You know that
An investor has \(\$ 10,000\) which she intends to invest in a portfolio of three stocks that she feels are good investment prospects. During the investor's planning horizon, the weekly closing prices of the stocks can be viewed as the outcome of a trivariate normal random variable
Let \(Y\) have a chi-square distribution with 15 degrees of freedom, let \(X\) have a chi-square distribution with 5 degrees of freedom, and let \(Y=X+Z\), where \(X\) and \(Z\) are independent random variables.(a) Calculate \(P(y>27.488)\).(b) Calculate \(P(6.262c)=.05\).
Let \(Y\) have the density \(N(5,36), X\) have the density \(N(4,25)\), let \(Y\) and \(X\) be independent random variables, and define \(W=X-Y\).(a) Calculate \(P(y>10)\).(b) Calculate \(P(-10c)=.95\).
Let \(\boldsymbol{X}\) be a bivariate random variable having the probability density \(N(\boldsymbol{\mu}, \boldsymbol{\Sigma})\), with\(\boldsymbol{\mu}=\left[\begin{array}{l}5 \\ 8\end{array}ight]\) and \(\boldsymbol{\Sigma}=\left[\begin{array}{rr}2 & -1 \\ -1 & 3\end{array}ight]\).(a) Define the
WAYSAFE, a large retail supermarket, has a standard inspection policy that determines whether a shipment of produce will be accepted or rejected. Specifically, they examine 5 percent of the objects in any shipment received, and, if no defective produce is found in any of the items examined, the
The FLAMES-ARE-US Co. manufactures butane cigarette lighters. Your top-of-the-line lighter, which has the brand name "SURE-FLAME," costs \(\$ 29.95\). As a promotional strategy, the SURE-FLAME lighter carries a guarantee that if it takes more than five attempts before the lighter actually lights,
An instructor in an introductory economics class has constructed a multiple-choice test for the mid-term examination. The test consists of 20 questions worth 5 points each. For each question, the instructor lists four possible answers, of which only one is correct. John Partytime, a student in the
The liquid crystal display in the new Extime brand of digital watches is such that the probability it continues to function for at least \(x\) hours before failure is constant (for any given choice of the number \(x\) ), regardless of how long the display has already been functioning. The expected
The Department of Transportation in a foreign county establishes gas-mileage standards that automobiles sold in must meet or else a "gas guzzler" tax is imposed on the sale of the offending types of automobile. For the "compact, four-door" class of automobiles, the target average gas mileage is 25
An appliance manufacturer is conducting a survey of consumer satisfaction with appliance purchases. All customers that have purchased one of the company's appliances within the last year will be mailed a customer satisfaction survey. The company is contemplating the proportion of surveys that
Customers arrive at the rate of four per minute at a large bank branch in downtown Seattle. In its advertising, the bank stresses that customers will receive service promptly with little or no waiting.(a) What is the probability that there will be more than 25 customers entering the bank in a 5
The accounts of the Excelsior company are being audited by an independent accounting firm. The company has 200 active accounts, of which 140 are current accounts, 45 are past due 60 or more days, and 15 accounts are delinquent. The accounting firm will randomly choose five different accounts in
The Stonebridge Tire Co. manufactures passenger car tires. The manufacturing process results in tires that are either first-quality tires, blemished tires, or defective tires. The proportions of the tires manufactured that fall in the three categories are \(.88, .09\), and .03 , respectively. The
KoShop, a large retail department store, has a standard inspection policy that determines whether a shipment of products will be accepted or rejected. Specifically, they examine a randomly chosen sample of 10 percent of the objects in any shipment received, and, if no defectives are found in any of
Customers arrive, on average, at the rate of two per minute at a bank in downtown Portland.(a) Define an appropriate probability model for the experiment of observing the number of customers entering the bank in a 5-minute period.(b) What is the probability that there will be more than 20 customers
The accounts of Pullman Plumbers, Inc. are being audited by an independent accounting firm. The company currently has 100 active accounts, of which 50 are current accounts, 35 are past due 60 or more days, and 15 accounts are delinquent. The accounting firm will randomly choose five different
An instructor in an introductory economics class has a true-false section on the mid-term examination that consists of 10 questions worth 3 points each. Jeff Nostudy, a student in the class, has not attended class regularly, and knows that he is really not prepared to take the exam. Nonetheless, he
In the board game of "Aggravation", the game begins with each player having their four game pieces in a "base" holding area and a player is unable to place one of her game pieces on the board for play until they role either a one or a six with a fair die.(a) Define a probability model that can be
The Central Processing Unit (CPU) in a laptop computer that your company manufactures is known to have the following service life characteristics:\(P(x>s+t \mid x>s)=P(x>t) \forall s\) and \(\mathrm{t}>0\)where outcomes, \(x\), of the random variable \(X\) measure the operating life of the CPU,
The LCD screen on a popular smartphone has the following operating life characteristic where outcomes, x, of the random variable X measure the operating life of the screen, measured in 100,000 hour increments, until failure of the screen. The engineers in your company tell you that (a) Define
Let (X1; . . . ;X20) be 20 iid N(2; 4) random variables. Also, define the random variables Zi Xi –2/2 for i 1; . . . ; 20. Answer the following questions relating to these random variables. (a) What are the values of E(Z) and var (Z?)? (b) What are the values of EZ) and var (Z)? (c)
The grades assigned to students taking a midterm in a large principles of economics class is assumed to be normally distributed with a mean of 75 and a standard deviation of 7. The Professor teaching the class, known to be a "stringent grader", has indicated that based on this distribution of
The production function for the number of toy robots that your company manufactures in a week, expressed in 100 's of robots, is represented as follows:\(Q=\mathbf{a}^{\prime} \mathbf{x}-\mathbf{x}^{\prime} \mathbf{B} \mathbf{x}+\varepsilon\), where \(x=\left[\begin{array}{l}l \\
The weekly average price, in dollars per gallon, and quantity sold of organic milk, measured in thousands of gallons, in a large west coast market in the fall is represented by the outcomes of a bivariate random variable \((P, Q)\) having a multivariate normal distribution, \(N(\boldsymbol{\mu},
Given current technology, the production of active matrix color screens for notebook computers is a difficult process that results in a significant proportion of defective screens being produced. At one company the daily proportion of defective \(9.5^{\prime \prime}\) and \(10.4^{\prime \prime}\)
Central limit theorems have important applications in the area of quality control. One such application concerns so-called control charts, and in particular, \(\bar{X}\) charts, which are used to monitor whether the variation in the calculated mean levels of some characteristics of a production
The lifetime of a certain computer chip that your company manufactures is characterized by the population distribution f(z;θ)=1θe−z/θI(0,∞)(z)f(z;θ)=1θe−z/θI(0,∞)(z), where zz is measured in thousands of hours. Let (X1,…,Xn)(X1,…,Xn) represent iid random variables with the
In each case below, the outcome of some function, T(X(n))T(X(n)), of nn iid random variables X(n)=(X1,…,Xn)X(n)=(X1,…,Xn) is being considered for providing an estimate of some function of parameters, q(θ)q(θ). Determine whether E(T(X(n)))=q(θ)E(T(X(n)))=q(θ),
The daily number of customers entering a large grocery store who purchase one or more dairy products is given by the outcome of a binomial random variable Xt with parameters p and nt for day t. The number of customers who enter the grocery store on any given day, nt, is itself an outcome of a
Let \(\left(X_{1}, \ldots, X_{n}ight)\) be iid random variables with \(\sigma^{2}
Let the random variables in the sequence \(\left\{X_{n}ight\}\) be iid with a gamma density having parameters \(\alpha=2\) and \(\beta=3\).(a) What is the probability density for \(\bar{X}_{n}\) ?(b) What is the asymptotic probability density for \(\bar{X}_{n}\) ?(c) Plot the actual versus
A pharmaceutical company claims that it has a drug that is 75 percent effective in generating hair growth on the scalps of balding men. In order to generate evidence regarding the claim, a consumer research agency conducts an experiment whereby a total of 1,000 men are randomly chosen and treated
A political candidate has hired a polling firm to assess her chances of winning a senatorial election in a large eastern state. She wants an estimate of the proportion of registered voters that would vote for her "if the election were held today." Registered voters are to be randomly chosen,
Let observations on the quantity supplied of a certain commodity be generated by \(Y_{i}=x_{i} \beta+V_{i}\), where \(\left|x_{i}ight| \in[a\), b] \(\forall i\) are scalar observations on fixed prices, \(\beta\) is an unknown slope coefficient, and the \(V_{i}^{\prime}\) 's are iid random variables
Let X1,…,XnX1,…,Xn be iid random variables having continuous uniform distributions of the form f(z)=I(0,1)(z)f(z)=I(0,1)(z). (a) Define an asymptotic distribution for ¯Xn=n−1∑ni=1XiX¯n=n−1∑i=1nXi. (b) Using your result from (a), argue that
A company produces a popular beverage product that is distributed nationwide. The aggregate demand for the product during a given time period can be represented by n n Q = Q = ( i B p + V) i=1 i=1 where Q, is quantity purchased by the ith consumer, i >0, i>0, E(V)=0, var(Vi)>c>0, P(|vi|
The daily tonnage of garbage handled by the EnviroSafe Landfill Co. is represented as the outcome of a random variable having some triangular distribution, asThis distribution is represented graphically as follows:Enviro-Safe is in the process of analyzing whether or not they need to expand their
A statistician wants to use iid outcomes from some exponential distribution f(x;θ)=(1/θ)e−x/θI(0,∞)(x)f(x;θ)=(1/θ)e−x/θI(0,∞)(x) to generate an estimate of the variance of the exponential density, θ2θ2. She wants to use the outcome of ¯X2nX¯n2 where
We have shown that if {Yn} is a sequence of w2 random variables, where Yn ~, then (Yn - n)/2nN(0, 1). Since Y we know that P(y2534.3816) Ply 50 63.1671) = P(y100118.498) = .90. Assign (approx- imate) probabilities to the three events using asymptotic distributions. How good are the approximations?
Let \(\left\{X_{n}ight\}\) be a sequence of random variables having binomial densities, where \(X_{n}\) has a binomial density with parameters \(n\) and \(p\), i.e.,\(X_{n} \sim\left(\begin{array}{c}n \\ x\end{array}ight) p^{x}(1-p)^{n-x} I_{\{0,1,2, \ldots n\}}(x)\).(a) Show that \(\left(X_{n}-n
The Nevada Gaming Commission has been directed to check the fairness of a roulette wheel used by the WINBIG Casino. In particular, a complaint was lodged stating that a "red" slot occurs more frequently than a "black" slot for the roulette wheel used by WINBIG, whereas red and black should occur
The Elephant Memory Chip Co. (EMC for short) instructs its resident statistician to investigate the operating-life characteristics of their new 4 gigabyte memory chip in order to provide product information to potential buyers. The population distribution of operating lives can be specified as some
Let (X1,…,Xn)(X1,…,Xn) be a random sample from a Poisson population distribution. Derive the limiting distribution of T=(¯Xn−μ)(S2n/n)1/2T=(X¯n−μ)(Sn2/n)1/2 where S2n=n−1∑ni=1(Xi−¯Xn)2Sn2=n−1∑i=1n(Xi−X¯n)2.
Liquid crystal displays (LCDs) that your wholesaling company is marketing for a large Japanese electronics firm are known to have a distribution of lifetimes of the following Gamma-distribution form:\(f(z ; \alpha)=\frac{1}{2^{\alpha} \Gamma(\alpha)} z^{\alpha-1} e^{-z / 2} I_{(0, \infty)}(z)\),
In each case below, determine whether the random variable sequence \(\left\{Y_{n}ight\}\) converges in probability and/or in mean square, and if so, define what is being converged to.(a) \(\quad Y_{j}=(j+5)^{-1} \sum_{i=1}^{j} X_{i} \quad\) for \(\quad j=1,2,3, \ldots\); \(X_{i}{ }^{\prime} S
In each case below, derive the probability limit and an asymptotic distribution for \(n^{-1} \sum_{i=1}^{n} X_{i}\) and a limiting distribution for the random variable \(Y_{n}\), if they can be defined.(a) \(X_{i}^{\prime}\) s iid Bernoulli \((p), Y_{n}=\frac{n^{-1} \sum_{i=1}^{n} X_{i}-p}{n^{-1 /
Let \(X\) be a random sample of size \(n\) from a \(N\left(\mu, \sigma^{2}ight)\) population distribution representing the weights, in ounces, of cereal placed in cereal boxes for a certain brand and type of breakfast cereal. Define \(\hat{\sigma}\) as in Theorem 6.19.(a) Show that the random
Let \(X\) and \(Y\) be two independent random samples of sizes \(n_{X}\) and \(n_{y}\), respectively, from two normal population distributions that do not necessarily have the same means or variances. The two distributions refer to the miles per gallon achieved by two \(1 / 2\)-ton pickup trucks
Let \(X\) be a random sample of size \(\mathrm{n}\) from a \(N\left(\mu, \sigma^{2}ight)\) population distribution representing the yield per acre, in pounds, of a new strain of hops used in the production of premium beer. (a) Justify that the random interval \(\left(n S^{2} / \chi_{\alpha}^{2}, n
The shipping and receiving department of a large toy manufacturer is contemplating two strategies for sampling incoming parts deliveries and estimating the proportion, \(p\), of defective parts in a shipment. The two strategies are differentiated on the basis of whether random sampling will be with
GenAG, Inc., a genetics engineering laboratory specializing in the production of better seed varieties for commercial agriculture, is analyzing the yield response to fertilizer application for a new variety of overlineley that it has developed. GenAg has planted 40 acres of the new overlineley
The Always Ready Battery Co. has developed a new "Failsafe" battery which incorporates a small secondary battery that becomes immediately functional upon failure of the main battery. The operating life of the main battery is a gamma distributed random variable as \(X_{1} \sim\) Gamma(3,1) where
The seasonal catch of a commercial fishing vessel in a certain fishery in the southern hemisphere can be represented by \(Q=c(\mathbf{z}) V\), where \(\mathbf{z}\) is a vector of characteristics of the vessel relating to tonnage, length, number of crew members, holding tank size, etc.,
A company markets its line of products directly to consumers through telephone solicitation. Salespersons are given a base pay that depends on the number of documented phone calls made plus incentive pay for each phone call that results in a sale. It can be assumed that the number of phone calls
The daily quantity of a commodity that can be produced using a certain type of production technology is given by the outcome of the random variable \(Q\), defined as \(Q=10 x_{1}^{35} x_{2}^{5} V\), where \(Q\) is measured in tons/day, \(x_{1}\) represents units of labor per day, \(x_{2}\)
The daily price, \(p\), and quantity sold, \(q\), of ground beef produced by the Red Meat Co. can be represented by outcomes of the bivariate random variable \((P, Q)\) having bivariate density function\(f(p, q)=2 p e^{-p q} I_{[.5,1]}(p) I_{(0, \infty)}(q)\)where \(p\) is measured in dollars and
Let \(X=\left(X_{1}, \ldots, X_{26}ight)\) and \(Y=\left(Y_{1}, \ldots, Y_{31}ight)\) represent two independent random samples from two normal population distributions. Let \(S_{X}^{2}\) and \(S_{Y}^{2}\) represent the sample variances associated with the two random samples and let \(\bar{X}\) and
The daily price, \(p\), and daily quantity sold, \(q\), of pink salmon produced by the AJAX Fish Packing Co. can be represented by outcomes of the bivariate random variable \((P, Q)\) with density function\(f(p, q)=5 p e^{-p q} I_{[.2,4]}(p) I_{(0, \infty)}(q)\)where \(p\) is measured in dollars,
The probability that a customer entering an electronics store will make a purchase is equal to \(p=.15\), and customers' decisions whether to purchase electronics equipment are jointly independent random variables.(a) Simulate the buying behavior of 10 customers entering the store using the
The number of times that a copy machine malfunctions in a day is the outcome of a Poisson process with \(\lambda=.1\).(a) Simulate the operating behavior of the copy machine regarding the daily number of malfunctions over a 10 day period using the following 10 outcomes from a Uniform \((0,1)\)
The length of time that a 4 gigabyte \(\mathrm{PC}\) computer memory module operates until failure is the outcome of an exponential random variable with mean \(\mathrm{E} X=3.25\), where \(x\) is measured in 100,000 hour units.(a) Simulate the operating lives of 10 memory modules using the
The monthly proportion of purchases paid by check to a large grocery store that are returned because of insufficient funds can be viewed as the outcome of a Beta \((1,20)\) distribution.(a) Simulate 12 monthly proportions of returned checks using the following 24 outcomes from a Uniform \((0,1)\)
The daily closing price for a certain stock issue on the NYSE can be represented as the outcome of \(Y_{t}=Y_{t-1}+V_{t}\), where \(y_{t}\) is the value of the stock price on day \(t\), and \(V_{t} \sim N(0,4)\) (This is an example of a stochastic process known as a random walk.)(a) Use the change
A random sample of the gas mileage achieved by 20 domestic compact automobiles resulted in the following outcome:(25.52,24.90,22.24,22.36,26.62,23.46,25.46,24.98,25.82, 26.10,21.59,22.89,27.82,22.40,23.98,27.77,23.29,24.57, 23.97,24.70).(a) Define and graph the empirical distribution function.(b)
The time between work-related injuries at the Imperial Tool and Die Co. during a given span of time resulted in the following 20 observations, where time was measured in weeks:\((9.68,6.97,7.08, .50,6.71,1.13,2.20,9.98,4.63,7.59,3.99,3.26\), \(.92,3.07,17.96,4.69,1.80,8.73,18.13,4.02)\).(a) Define
A realtor randomly samples homeowners who have purchased homes in the last 2 years and records their income, \(y\), and home purchase price, \(p\) (the population is large enough that one can consider this a random sample with replacement):(a) Calculate the sample covariance between income and home
The proportion of the work force of a large Detroit manufacturing firm that takes at least 1 day's sick leave in a given work week is assumed to be the outcome of a random variable whose PDF is well-represented by a uniform distribution on the interval \([0, .10]\).(a) In a random sample of eight
A large aircraft manufacturer produces a passenger jet having a navigation component consisting of three sequentially functioning redundant navigation systems that will allow the jet to be properly controlled so long as at least one of the systems remain operational. The operating life of each of
A news agency wants to poll the population of registered voters in the United States (over 200,000,000) to find out how many would vote for the Republican candidate, Henry Washington, if the election were held today. They intend to take a random sample, with replacement, of 1,000 registered U.S.
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