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Questions and Answers of Statistics

Many manufacturing companies in the United States and abroad use molded parts as components of a process. Shrinkage is often a major problem. Thus a molded die for a part is built larger than nominal
Consider the situation of Exercise 1.28. But now use the following data set in which shrinkage is measured once again at low injection velocity and high injection velocity. However, this time the
Use the results of Exercises 1.28 and 1.29 to create a plot that illustrates the interaction evident from the data. Use the plot in Figure in Example 1.3 as a guide. Could the type of information
Assume that two random variables (X, V) arc uniformly distributed on a circle with radius a. Then the joint probability density function is Find the expected value of X,fix-
The probability distribution of the discrete random variable X is find the mean ofX.
Find the mean of the random variable T representing the total of the three coins of Exercise 3.25.
A coin is biased so that a bead is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.
The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given in Exercise 3.13 as Find the average number of
An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are 1/12, 1/12, 1/4, 1/4, 1/6, and 1/6, respectively, that the attendant receives $7,
By investing in a particular stock, a person can make a profit in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person's expected gain?
Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0.22, 0.36, 0.28, and 0.14 respectively, that she will be able to sell it, for a
In a gambling game a woman is paid $3 if she draws a jack or a queen and $5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much
Two tire-quality experts examine stacks of tires and assign quality ratings to each tire on a 3-point scale. Let X denote the grade given by expert A and Y denote the grade given by B. The following
A private pilot wishes to insure his airplane for $200,000. The insurance company estimates that a total loss may occur with probability 0.002, a 50% loss with probability 0.01, and a 25% loss with
If a dealer's profit, in units of $5000, on a new automobile can bo looked upon as a random variable X having the density function find the average profit perautomobile.
The density function of coded measurements of pitch diameter of threads of a fitting is Find the expected value ofX.
What, proportion of individuals can be expected to respond to a certain mail-order solicitation if the proportion X has the density function
The density function of the continuous random variable X, the total number of hours, in units of 100 hours that a family runs a vacuum cleaner over a period of one year, is given in Exercise 3.7.
Let X be a random variable with the following probability distribution: Find s(x) where g (X) = (2X +1)2,
Find the expected value of the random variable g(X) = X2, where X has the probability distribution of Exercise 4.2.
A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous year. Suppose that the number of word
A continuous random variable X has the density function Find the expected value of g(X) = e2X/3.
What is the dealer's average profit per automobile if the profit on each automobile is given by g(X) = X2, where X is a random variable having the density function of Exercise 4.12?
The hospital period, in days, for patients following treatment for a certain typo of kidney disorder is a random variable Y = X + 4, where X has the density function Find the average number of days
Suppose that X and Y have the following joint probability function:(a) Find the expected value of g(X, Y) = XY2.(b) Find ?x and ?y.
Referring to the random variables whose joint probability distribution is given in Exercise 3.39. (a) Find E (X2Y – 2XY); (b) Find μx – μy.
Referring to the random variables whose joint probability distribution is given in Exercise 3.53 find the mean for the total number of jacks and kings when 3 cards are drawn without replacement from
Let X and Y be random variables with joint density function Find the expected value of Z = √/X2 + Y2.
In Exercise 3.27, a density function is given for the time to failure of an important component of a DVD player. Find the mean number of hours to failure of the component and thus the DVD.
Consider the information of Exercise 3.28. The problem deals with the weight in ounces of the product in a, cereal box with(a) Plot, the density function.(b) Compute the expected value or mean weight
In Exercise 3.29, we were dealing with an important particle size distribution with the distribution of the particle size characterized by(a) Plot the density function.(b) Give the mean particlesize.
In Exercise 3.31, the distribution of time before a major repair of a washing machine was given as what is the population mean "time torepair?"
Consider Exercise 3.32.(a) What is the mean proportion of the budget, allocated to environmental and pollution control?(b) What is the probability that a company selected at random will have a
In Exercise 3.13, the distribution of the number of imperfections per 10 meters of synthetic fabric is given by (a) Plot the probability function. (b) Find the expected number of imperfections E(X) =
Use Definition 4.3 to find the variance of the random variable X of Exercise 4.7.
Let X be a random variable with the following probability distribution: Find the standard deviation ofX.
The random variable X, representing the number of errors per 100 lines of software code, has the following probabilitydistribution:
Suppose that the probabilities are 0.4. 0.3, 0.2, and 0.1, respectively, that 0, 1, 2 or 3 power failures will strike a certain subdivision in any given year. Find the mean and variance of the random
A dealer's profit, in units of S5000, on a new automobile is a random variable X having the density function given in Exercise 4.12. Find the variance of A".
The proportion of people who respond to a certain mail-order solicitation is a random variable X having the density function given in Exercise 4.14. Find the variance of X.
The total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a random variable X having the density function given in Exercise 4.15. Find the
Referring to Exercise 4.14 find σ2g(X) for the function g(X) = 3A2 + 4.
Find the standard deviation of the random variable g(X) = (2X + l)2 in Exercise 4.17.Data from 4.17.Let X be a random variable with the following probability distribution: Find μg(x), where
Using the results of Exercise 4.21, find the variance of g(X) = X2, where X is a random variable having the density function given in Exercise:
The length of time, in minutes, for an airplane to obtain clearance for take off at a certain airport is a random variable Y = 3X ? 2, where X has the density function Find the mean and variance of
Find the covariance of the random variables A' and Y of Exercise 3.39.
Find the covariance of the random variables X and Y of Exercise 3.49.
Find the covariance of the random variables A' and Y of Exercise 3.44.
Referring to the random variables whose joint density function is given in Exercise 3.40, find the covariance of X and Y.
Given a random variable A, with standard deviation σx and a random variable Y = a + bX, show that if b < 0, the correlation coefficient pXY = —1, and if b > 0, pXY = 1.
Consider the situation in Exercise 4.32. The distribution of the number of imperfections per 10 meters of synthetic failure is given by Find the variance and standard deviation of the number
On a laboratory assignment, if the equipment is working, the density function of the observed outcome, X is Find the variance and standard deviation ofX.
Referring to Exercise 4.35 find the mean and variance of the discrete random variable Z = 3X — 2, when X represents the number of errors per 100 lines of code.
Using Theorems 4.5 and 4.9, find the mean and variance of the random variable Z = 5A + 3, where X has the probability distribution of Exercise 4.30.
Suppose that a grocery store purchases 5 cartons of skim milk at the wholesale price of $1.20 per carton and retails the milk at $1.65 per carton. After the expiration date, the unsold milk is
Repeat Exercise 4.43 on page 122 by applying Theorems 4.5 and 4.9.
Let X be a random variable with the following probability distribution: Find E(X) and E(X2) and then, evaluate E [(2X +1)2].
The total time, measured in units of 100 hours, that a teenager runs her stereo set over a period of one year is a continuous random variable A that has the density function Use Theorem 4.6 to
If a random variable X is defined such that find p, and σ2.
Suppose that X and Y are independent random variables having the joint probability distribution Find (a) E (2X ? 3Y); (b) E (XY).
Use Theorem 4.7 to evaluate E (2XY2 – X2Y) for the joint probability distribution shown in Table 3.1.
Seventy new jobs are opening up at an automobile manufacturing plant, but 1000 applicants show up for the 70 positions. To select the best 70 from among the applicants, the company gives a test that
An electrical firm manufactures a 100-watt light bulb, which, according to specifications written on the package, has a mean life of 900 hours with a standard deviation of 50 hours. At most, what
A local company manufactures telephone wire. The average length of the wire is 52 inches with a standard deviation of 6.5 inches. At most, what percentage of the: telephone wire from this company
Suppose that you roll a fair 10-sided die (0, 1, 2, . . . , 9) 500 times. Using Chebyshev's theorem, compute the probability that the sample mean, X, is between 4 and 5.
If X and Y are independent random variables with variances σ2x = 5 and at = 3, find the variance of the random variable Z = – 2X + 4Y – 3.
Repeat Exercise 4.64 if X and Y are not independent and σXY = 1.
A random variable X has a mean p = 12, a variance a2 = 9, and an unknown probability distribution. Using Chebyshev's theorem, estimate(a) P(6 < X < 18);(b) P(3 < X < 21).
A random variable X has a mean p = 10 and a variance σ2 = 4. Using Chebyshev's theorem, find. (a) P(X – 10| > 3);. (b) P (| X – 10 | < 3);. (c) P(b < X < 15);. (d) The value of the
Compute P(μ – 2σ < X < μ + 2a), where X has the density function and compare with the result given in Chebyshev’s theorem.
Let X represents the number that occur when a red die is tossed and Y the number that occurs when a green die is tossed. Find(a) E(X + Y);(b) E(X – Y);(c) E(XY).
Suppose that X and Y are independent random variables with probability densities and Find the expected value of Z =XY
If the joint density function of X and Y is given by find the expected value of g(X, y) = x/Y3 +X2Y.
Let X represents the number that occurs when a green die is tossed and Y the number that occurs when a red die is tossed. Find the variance of the random variable(a) 2X – Y;(b) X + 3Y – 5.
Consider a random variable X with density function (a) Find p = E(X) and σ2 = E[(X - p)2]. (b) Demonstrate that Chebyshev's theorem holds for k = 2 and k = 3.
The power P in watts which is dissipated in an electric circuit with resistance R is known to be given by P = I2 R, where I is current in amperes and R is a fixed constant at 50 ohms. However, I is a
Consider Review Exercise 3.79. The random variables X and V represent the number of vehicles that arrive at two separate street corners during a certain 2-minute period in the day. The joint
Consider Review Exercise 3.66. There are two service lines. The random variables X and Y are the proportion of time that line 1 and line 2 are in use respectively. The joint probability density
The length of time Y in minutes required to generate a human reflex to tear gas has density function(a) What is the mean time to reflex?(b) Find E(Y2) andVar(Y).
A manufacturing company has developed a fuel efficient machine for cleaning carpet because it delivers carpet cleaner so rapidly. Of interest is a random variable Y, the amount in gallons per minute
For the situation in Exercise 4.78, compute E(eY) using Theorem 4.1, that is, by using Then compute E(eY) by not using f(y), but rather by using the second-order adjustment to the first-order
Consider again the situation of Exercise 4.78. It is required to find Var (eY). Use Theorems 4.2 and 4.3 and define Z = eY. Thus, use the conditions of Exercise 4.79 to find Var (Z) = E (Z2) –
Prove Chebyshev's theorem when X is a discrete random variable.
Find the covariance of the random variables X and Y having the joint, probability densityfunction.
Referring to the random variables whose joint probability density function is given in Exercise 3.47 find the average amount of kerosene left in the tank at the end of the day.
Assume the length X in minutes of a particular type of telephone conversation is a random variable with probability density function(a) Determine the mean length JE(X) of this type of telephone
Referring to the random variables whose joint density function is given in Exercise 3.41, find the covariance between the weight of the creams and the weight of the toffees in these boxes of
Referring to the random variables whose joint probability density function is given in Exercise 3.41 find the expected weight for the sum of the creams and toffees if one purchased a box of these
Suppose it is known that the life X of a particular compressor in hours has the density function(a) Find the mean life of the compressor.(b) Find E(X2).(c) Find the variance and standard deviation of
Referring to the random variables whose joint density function is given in Exercise 3.40 on page 101, (a) Find μx and μy; (b) Find E [(X + V)/2],
Show that Cov(aX, bY) = ab Cov(X,V).
Consider the density function of the Review Exercise 4.87. Demonstrate that Chebyshev's theorem holds for k. = 2 and k = 3.
Consider the joint density function Compute the correlation coefficientpXY.2,0
Consider random variables X and Y of Exercise 4.65. Compute pXY.
A dealer's profit in units of $5000 on a new automobile is a random variable A" having density function(a) Find the variance of the dealer's profit.(b) Demonstrate that Chebyshev's inequality holds
Consider Exercise 4.10 can it, be said that the grades given by the two experts are independent? Explain why or why not.
A company's marketing and accounting departments have determined that if the company markets its newly developed product, the contribution of the product to the firm's profit during the next 6 months
An important system acts in support of a vehicle in our space program. A single crucial component works only 85% of the time. In order to enhance the reliability of the system, it is decided that 3
In business it is important to plan and carry on research in order to anticipate what will occur at the end of the year. Research suggests that the profit (loss) spectrum is as follows with
It is known through data collection and considerable research that the amount of time in seconds that. a certain employee of a company is late for work is a random variable X with density function In

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