Suppose that X has a lognormal distribution with parameters θ = 0 and ω2 = 4. Determine the following: a. P(10 < X < 50) b. Value for x such that P(X < x) = 0.05 c. Mean and variance of X
> Determine whether the following transactions are taxable. If a transaction is not taxable, indicate what type of reorganization is effected, if any. a. Alpha Corporation owns assets valued at $400,000 and liabilities of $100,000. Beta Corporation transfe
> Krol Corp. distributed marketable securities in redemption of its stock in a complete liquidation. On the date of distribution, these securities had a basis of $100,000 and a fair market value of $150,000. What gain does Krol have as a result of the dist
> On December 31 of the current year, after receipt of his share of partnership income, Fox sold his interest in a limited partnership for $50,000 cash plus relief of all liabilities. On that date, the adjusted basis of Fox's partnership interest was $60,0
> Desi's adjusted basis of her partnership interest was $40,000 immediately before she received a distribution in full liquidation of her Makris partnership interest. The distribution consisted of $25,000 in cash and land with a fair market value of $30,00
> Gearty's adjusted basis in Worthington Company, a partnership, was $ 18,000 at the time Gearty received the following non-liquidating distributions of partnership property: What is Gearty's tax basis in the land received from the partnership? a. SO b.
> Which of the following statements is not true? a. Affiliated corporations that file consolidated returns can take a l00% dividends received deduction. b. The dividends received deduction for a small investment in an unrelated corporation is 500A.. c. The
> Assume the same facts as in Problems 46 to 49. Assume also the following: (I) Amy's ordinary business income from AM is qualified business income; (2) Amy's taxable income excluding capital gains is $500,000; and (3) Amy has no income from REITs, publicl
> Assume the same facts as Problem 31, except that TAV distributes a $50,000 (FMV) interest in the land and $50,000 (FMV) of accounts receivable to Vincent and $25,000 of cash and $25,000 (FMV) of accounts receivable each to Anita and Tyler. In general ter
> Vincent is a 50% partner in the TAV Partnership. He became a partner three years ago when he contributed land with a value of $60,000 and a basis of $30,000 (current value is $100,000). Tyler and Anita each contributed $30,000 cash for a 25% interest. Vi
> Beach Corporation, a seaside restaurant, is owned by 20 unrelated shareholders. This year, Beach made the national news for polluting the surrounding beach and ocean with garbage; it was fined $200,000. Beach fired its chief executives and appointed a ne
> The Rho Corporation was incorporated eight years ago by Tyee and Danette. Tyee received 5,000 shares of common stock for his $100,000 contribution, and Danette received 10,000 shares of con1mon stock for her $200,000 contribution. Five years ago, both Ty
> Certain Corporation is the parent entity in a Federal consolidated group for corporate income tax purposes. When its wholly owned subsidiary, Likely, Inc., reports an operating profit, Certain's basis in the Likely stock increases, as does the balance in
> Tammy Olsen has owned l00% of the common stock of Green Corporation (basis of $75,000) since the corporation's formation in 2008. In 2015, when Green had E & P of $320,000, the corporation distributed to Tammy a nontaxable dividend of 500 shares of prefe
> Suppose that X has the probability distribution fX(x) = 1, 1 ≤ x ≤ 2 Determine the probability distribution of the random variable Y = eX.
> The velocity of a particle in a gas is a random variable V with probability distribution fV (v) = av2e−bv , v > 0 where b is a constant that depends on the temperature of the gas and the mass of the particle. a. Determine the value of the constant a. b.
> A random variable X has the probability distribution fX(x) = e−x , x ≥ 0 Determine the probability distribution for the following: a. Y = X2 b. Y = X1/2 c. Y = ln X
> Suppose that X is a continuous random variable with probability distribution fX(x) = x /18 , 0 ≤ x ≤ 6 a. Determine the probability distribution of the random variable Y = 2X + 10. b. Determine the expected value of Y.
> Suppose that X is a random variable with probability distribution fX(x) = 1∕4, x = 1, 2, 3, 4 Determine the probability distribution of Y = 2X + 1.
> The probability density function of the net weight in pounds of a packaged chemical herbicide is f (x) = 2.0 for 49.75 < x < 50.25 pounds. a. Determine the probability that a package weighs more than 50 pounds. b. How much chemical is contained in 90% of
> Suppose that f (x) = 1.5x2 for −1 < x < 1. Determine the following: a. P(0 < X) b. P(0.5 < X) c. P(−0.5 ≤ X ≤ 0.5) d. P(X < −2) e. P(X < 0 or X > −0.5) f. x such that P(x < X) = 0.05.
> Suppose that f (x) = x∕8 for 3 < x < 5. Determine the following probabilities: a. P(X < 4) b. P(X > 3.5) c. P(4 < X < 5) d. P(X < 4.5) e. P(X < 3.5 or X > 4.5)
> Suppose that f (x) = 0.5 cos x for −π∕2 < x < π∕2. Determine the following: a. P(X < 0) b. P(X < −π∕4) c. P(−π∕4 < X < π∕4) d. P(X > −π∕4) e. x such that P(X < x) = 0.95
> The diameter of a particle of contamination (in micrometers) is modeled with the probability density function f (x) = 2∕x3 for x > 1. Determine the following: a. P(X < 2) b. P(X > 5) c. P(4 < X < 8) d. P(X < 4 or X > 8) e. x such that P(X < x) = 0.95
> Suppose that Xi has a normal distribution with mean μi and variance σ2i , i = 1, 2. Let X1 and X2 be independent. a. Find the moment-generating function of Y = X1 + X2. b. What is the distribution of the random variable Y?
> The distribution of X is approximated with a triangular probability density function f (x) = 0.0025x − 0.075 for 30 < x < 50 and f (x) = −0.0025x + 0.175 for 50 < x < 70. Determine the following: a. P(X ≤ 40) b. P(40 < X ≤ 60) c. Value x exceeded with p
> Suppose that f (x) = e−x for 0 < x. Determine the following: a. P(1 < X) b. P(1 < X < 2.5) c. P(X = 3) d. P(X < 4) e. P(3 ≤ X) f. x such that P(x < X) = 0.10 g. x such that P(X ≤ x) = 0.10
> Determine the value of c that makes the function f (x, y) = ce−2x−3y a joint probability density function over the range 0 < x and x < y. Determine the following: a. P(X < 1, Y < 2) b. P(1 < X < 2) c. P(Y > 3) d. P(X < 2, Y < 2) e. E(X) f. E(Y) g. Mar
> Determine the value of c such that the function f (x, y) = cxy for 0 < x < 3 and 0 < y < 3 satisfies the properties of a joint probability density function. Determine the following: a. P(X < 2, Y < 3) b. P(X < 2.5) c. P(1 < Y < 2.5) d. P(X > 1.8, 1 < Y
> An article in Electric Power Systems Research [“Modeling Real-Time Balancing Power Demands in Wind Power Systems Using Stochastic Differential Equations” (2010, Vol. 80(8), pp. 966–974)] considered a
> An engineering statistics class has 40 students; 60% are electrical engineering majors, 10% are industrial engineering majors, and 30% are mechanical engineering majors. A sample of four students is selected randomly without replacement for a project tea
> An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the TPC-C OLTP (Transaction Processing Performance Council’s Version C On-Lin
> A manufacturing company employs two devices to inspect output for quality control purposes. The first device is able to accurately detect 99.3% of the defective items it receives, whereas the second is able to do so in 99.7% of the cases. Assume that fou
> A small-business Web site contains 100 pages and 60%, 30%, and 10% of the pages contain low, moderate, and high graphic content, respectively. A sample of four pages is selected randomly without replacement, and X and Y denote the number of pages in the
> In the transmission of digital information, the probability that a bit has high, moderate, and low distortion is 0.01, 0.04, and 0.95, respectively. Suppose that three bits are transmitted and that the amount of distortion of each bit is assumed to be in
> Let X1, X2,…, Xr be independent exponential random variables with parameter λ. a. Find the moment-generating function of Y = X1 + X2 + …+Xr. b. What is the distribution of the random variable Y?
> Show that the following function satisfies the properties of a joint probability mass function. Determine the following: a. P(X b. P(X c. P(Y d. P(X > 0.25, Y e. E(X), E(Y), V(X), V(Y) f. Marginal probability distribution of X
> Show that the following function satisfies the properties of a joint probability mass function. Determine the following: a. P(X b. P(X c. P(Y d. P(X > 1.8, Y > 4.7) e. E(X), E(Y), V(X), V(Y) f. Marginal probability distribution of X
> Solve the following
> F(x) = 1 − e−2x x > 0
> Determine the cumulative distribution function for the distribution in Exercise 4.1.7. Use the cumulative distribution function to determine the probability that a length exceeds 2.7 meters.
> An article in IEEE Journal on Selected Areas in Communications [“Impulse Response Modeling of Indoor Radio Propagation Channels” (1993, Vol. 11(7), pp. 967–978)] indicated that the successful design of indoor communication systems requires characterizati
> Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to uncertainties in financial resources. The proportion of completion within one year has a beta distribution with parameters α = 1 an
> An airline makes 200 reservations for a flight that holds 185 passengers. The probability that a passenger arrives for the flight is 0.9, and the passengers are assumed to be independent. a. Approximate the probability that all the passengers who arrive
> A square inch of carpeting contains 50 carpet fibers. The probability of a damaged fiber is 0.0001. Assume that the damaged fibers occur independently. a. Approximate the probability of one or more damaged fibers in one square yard of carpeting. b. Appro
> The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch. a. Suppose that the specifications require the dot diameter to be between 0.0014 and 0.0026 inch. If the probability that a dot meets specifications
> A random variable X has the gamma distribution a. Show that the moment-generating function for t b. Find the mean and variance of X.
> The waiting time for service at a hospital emergency department follows an exponential distribution with a mean of three hours. Determine the following: a. Waiting time is greater than four hours b. Waiting time is greater than six hours given that you h
> Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per square centimeter of surface dust is 100. A sample of 800
> Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively. Determine the following: a. Parameters θ and ω2 of the lognormal distribution b. Probability that X is less than 150
> Suppose that f (x) = 0.5x − 1 for 2 < x < 4. Determine the following: a. P(X < 2.5) b. P(X > 3) c. P(2.5 < X < 3.5) d. Determine the cumulative distribution function of therandom variable. e. Determine the mean and variance of the random variable.
> The time between calls is exponentially distributed with a mean time between calls of 10 minutes. a. What is the probability that the time until the first call is less than five minutes? b. What is the probability that the time until the first call is be
> The life of a recirculating pump follows a Weibull distribution with parameters β = 2 and δ = 700 hours. Determine for parts (a) and (b): a. Mean life of a pump b. Variance of the life of a pump c. What is the probability that a pump will last longer tha
> The size of silver particles in a photographic emulsion is known to have a log normal distribution with a mean of 0.001mm and a standard deviation of 0.002 mm. a. Determine the parameter values for the lognormal distribution. b. What is the probability o
> When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that eac
> The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with λ = 0.00004. What is the probability that the time until failure is a. At least 20,000 hours? b. At most 30,000 hours? c. Between 20,000 and
> The percentage of people exposed to a bacteria who become ill is 20%. Assume that people are independent. Assume that 1000 people are exposed to the bacteria. Approximate each of the following: a. Probability that more than 225 become ill b. Probability
> An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Acute Myocardial Infarction Patients Using Non-Parametric and Parametric Approaches” (2009, Vol. 2(1), pp. 22–36)] described the use of aWeibull distribution to model
> An article in Electric Power Systems Research [“On the Self-Scheduling of a Power Producer in Uncertain Trading Environments” (2008, Vol. 78(3), pp. 311–317)] considered a self-scheduling approach for a power producer. In addition to price and forced out
> An article in Journal of Theoretical Biology [“Computer Model of Growth Cone Behavior and Neuronal Morphogenesis” (1995, Vol. 174(4), pp. 381–389)] developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in
> Among homeowners in a metropolitan area, 25% recycle paper each week. A waste management company services 10,000 homeowners (assumed independent). Approximate the following probabilities: a. More than 2600 recycle paper in a week b. Between 2400 and 2600
> Provide approximate sketches for beta probability density functions with the following parameters. Comment on any symmetries and show any peaks in the probability density functions in the sketches. a. α = β < 1 b. α = β = 1 c. α = β > 1
> Consider the regional right ventricle transverse wall motion in patients with pulmonary hypertension (PH). The right-ventricle ejection fraction (EF) is approximately normally distributed with a standard deviation of 12 for PH subjects, and with mean and
> The intensity (mW/mm2) of a laser beam on a surface theoretically follows a bivariate normal distribution with maximum intensity at the center, equal variance σ in the x and y directions, and zero covariance. There are several definitions for the width o
> The power in a DC circuit is P = I2/R where I and R denote the current and resistance, respectively. Suppose that I is approximately normally distributed with mean of 200 mA and standard deviation 0.2 mA and R is a constant. Determine the probability den
> Suppose X has a lognormal distribution with parameters θ and ω. Determine the probability density function and the parameters values for Y = Xγ for a constant γ > 0. What is the name of this distribution?
> The continuous uniform random variable X has density function a. Show that the moment-generating function is b. Use MX(t) to find the mean and variance of X.
> Amarketing company performed a risk analysis for a manufacturer of synthetic fibers and concluded that new competitors present no risk 13% of the time (due mostly to the diversity of fibers manufactured), moderate risk 72% of the time (some overlapping o
> The permeability of a membrane used as a moisture barrier in a biological application depends on the thickness of two integrated layers. The layers are normally distributed with means of 0.5 and 1 millimeters, respectively. The standard deviations of lay
> A mechanical assembly used in an automobile engine contains four major components. The weights of the components are independent and normally distributed with the following means and standard deviations (in ounces): a. What is the probability that the we
> Suppose that X and Y have a bivariate normal distribution with σX = 4, σY = 1, μX = 2, μY = 4, and ρ = −0.2. Draw a rough contour plot of the joint probability density function.
> The weight of a small candy is normally distributed with a mean of 0.1 ounce and a standard deviation of 0.01 ounce. Suppose that 16 candies are placed in a package and that the weights are independent. a. What are the mean and variance of the package’s
> The time for an automated system in a warehouse to locate a part is normally distributed with a mean of 45 seconds and a standard deviation of 30 seconds. Suppose that independent requests are made for 10 parts. a. What is the probability that the averag
> Contamination problems in semiconductormanufacturing can result in a functional defect, a minor defect, or no defect in the final product. Suppose that 20%, 50%, and 30% of the contamination problems result in functional, minor, and no defects, respectiv
> The joint distribution of the continuous random variables X, Y, and Z is constant over the region x2 + y2 ≤ 1, 0 < z < 4. Determine the following: a. P(X2 + Y2 ≤ 0.5) b. P(X2 + Y2 ≤ 0.5, Z < 2) c. Joint conditional probability density function of X and
> A continuous random variable X has the following probability distribution: f (x) = 4xe−2x , x > 0 a. Find the moment-generating function for X. b. Find the mean and variance of X.
> Determine the value of c such that the function f (x, y) = cx2y for 0 < x < 3 and 0 < y < 2 satisfies the properties of a joint probability density function. Determine the following: a. P(X < 1, Y < 1) b. P(X < 2.5) c. P(1 < Y < 2.5) d. P(X > 2.1 < Y <
> The percentage of people given an antirheumatoid medication who suffer severe, moderate, or minor side effects are 10, 20, and 70%, respectively. Assume that people react independently and that 20 people are given the medication. Determine the following:
> Show that the following function satisfies the properties of a joint probability mass function: Determine the following: a. P(X c. P(X 0.5, Y e. E(X), E(Y), V(X), V(Y) f. Marginal probability distribution of the random variable X g. Conditional probabil
> The sick-leave time of employees in a firmin a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours. a. What is the probability that the sick-leave time for next month will be between 50 and 80 hours? b. How much ti
> The time it takes a cell to divide (called mitosis) is normally distributed with an average time of 1 hour and a standard deviation of 5 minutes. a. What is the probability that a cell divides in less than 45 minutes? b. What is the probability that it t
> The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is f (x) = 0.04 for 50 < x < 75 seconds. a. What percentage of tests requires more than 70 seconds to complete? b. What percentage of tes
> The probability density function of the time you arrive at a terminal (in minutes after 8:00 A.M.) is f (x) = 0.1 exp(−0.1x) for 0 < x. Determine the probability that a. You arrive by 9:00 A.M. b. You arrive between 8:15 A.M. and 8:30 A.M. c. You arrive
> Determine the cumulative distribution function for the distribution in Exercise 4.1.4.
> Determine the cumulative distribution function for the distribution in Exercise 4.1.2.
> An electron emitter produces electron beams with changing kinetic energy that is uniformly distributed between 3 and 7 joules. Suppose that it is possible to adjust the upper limit of the kinetic energy (currently set to 7 joules). a. What is the mean ki
> The chi-squared random variable with k degrees of freedom has moment-generating function MX(t) = (1 − 2t)−k∕2. Suppose that X1 and X2 are independent chi-squared random variableswith k1 and k2 degrees of freedom, respectively. What is the distribution of
> An e-mail message will arrive at a time uniformly distributed between 9:00 A.M. and 11:00 A.M. You check e-mail at 9:15 A.M. and every 30 minutes afterward. a. What is the standard deviation of arrival time (in minutes)? b. What is the probability that t
> The volume of a shampoo filled into a container is uniformly distributed between 374 and 380 milliliters. a. What are the mean and standard deviation of the volume of shampoo? b. What is the probability that the container is filled with less than the adv
> A show is scheduled to start at 9:00 A.M., 9:30 A.M., and 10:00 A.M. Once the show starts, the gate will be closed. A visitor will arrive at the gate at a time uniformly distributed between 8:30 A.M. and 10:00 A.M. Determine the following: a. Cumulative
> An adult can lose or gain two pounds of water in the course of a day. Assume that the changes in water weight are uniformly distributed between minus two and plus two pounds in a day.What is the standard deviation of a person’s weight over a day?
> The thickness of photoresist applied towafers in semiconductor manufacturing at a particular location on the wafer is uniformly distributed between 0.2050 and 0.2150 micrometers. Determine the following: a. Cumulative distribution function of photoresist
> The thickness of a flange on an aircraft component is uniformly distributed between 0.95 and 1.05 millimeters. Determine the following: a. Cumulative distribution function of flange thickness b. Proportion of flanges that exceeds 1.02 millimeters c. Thic
> A random variable X has the discrete uniform distribution a. Show that the moment-generating function is b. Use MX(t) to find the mean and variance of X.
> Determine the covariance and correlation for the CD4 counts in a month and the following month in Exercise 5.2.6.
> The joint probability distribution is Show that the correlation between X and Y is zero but X and Y are not independent.
> Determine the covariance and correlation for the joint probability density function fXY (x, y) = e−x−y over the range 0 < x and 0 < y.
> Determine the value for c and the covariance and correlation for the joint probability density function fXY (x, y) = cxy over the range 0 < x < 3 and 0 < y < x.
> Assume that Z has a standard normal distribution. Use Appendix Table III to determine the value for z that solves each of the following: a. P(−z < Z < z) = 0.95 b. P(−z < Z < z) = 0.99 c. P(−z < Z < z) = 0.68 d. P(−z < Z < z) = 0.9973
> Use Appendix Table III to determine the following probabilities for the standard normal random variable Z: a. P(Z < 1.32) b. P(Z < 3.0) c. P(Z > 1.45) d. P(Z > −2.15) e. P(−2.34 < Z < 1.76)
> Suppose X has a continuous uniformdistribution over the interval [−1, 1]. Determine the following: a. Mean, variance, and standard deviation of X b. Value for x such that P(−x < X < x) = 0.90 c. Cumulative distribution function
