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Probability And Statistics For Engineers And Scientists 3rd Edition Anthony Hayter - Solutions

The random variable X measures the concentration of ethanol in a chemical solution, and the random variable Y measures the acidity of the solution. They have a joint probability density function f(x, y) = A(20-x-2y) for 0 < x < 5 and 0 < y < 5, and f(x, y) = 0 elsewhere. (a) What is the value of
Two safety inspectors inspect a new building and assign it a "safety score" of 1, 2, 3, or 4. Suppose that the random variable X is the score assigned by the first inspector and the random variable Y is the score assigned by the second inspector, and that they have a joint probability mass function
Suppose that the random variables X, Y and Z are independent with E(X) = 2, Var(X) = 4. E(Y) = -3, Var(K) = 2. E(Z) = 8 and Var(Z) = 7. Calculate the expectation and variance of the following random variables. (a) 3X + 7 (b) 5X - 9 (c) 2A + 6K (d) 4X - 3Y (e) 5X - 9Z + 8 (f) -3Y - Z - 5 (g) X + 2K
A rod of length L is bent until it snaps in two. The point of breakage X, as measured from one end of the rod, has a probability density function f(x) = Ax(L - x) for 0 < x < L. (a) What is the value of A? (b) What is the probability density function of the difference in the lengths of the two
If $x is invested in mutual fund A. the annual return has an expectation of $0.lx and a standard deviation of S0.02.V. If $x is invested in mutual fund B. the annual return has an expectation of $0.lx and a standard deviation of $0.03.1x. Suppose that the returns on the two funds are independent of
Recall Problems 2.2.11, 2.3.17, and 2.4.13 concerning the resistance of an electrical component. Suppose that five of the components are used in an electrical circuit so that the total resistance is the sum of the five individual resistances. Furthermore, suppose that it is reasonable to assume
Suppose that items from a manufacturing process are subject to three separate evaluations, and that the results of the first evaluation X1 have a mean value of 59 with a standard deviation of 10, the results of the second evaluation X2 have a mean value of 67 with a standard deviation of 13, and
The random variable X has an expectation of 77 and a standard deviation of 9. Find the values of a and b such that the random variable Y = a + bX has an expectation of 1000 and a standard deviation of 10.
Suppose that components are manufactured such that their heights are independent of each other with μ = 65.90 and σ = 0.32. (a) What are the mean and the standard deviation of the average height of five components? (b) If eight components are stacked on top of each other, what are the mean and
An object has a weight W. When it is weighed with machine 1, a value X1 is obtained. When it is weighed with machine 2, a value X2 is obtained. The value X1 has a mean IV and a standard deviation 3. The value X2 has a mean W and a standard deviation 4. The values X1 and X2 are independent. (a)
A fair six-sided die is rolled 80 times and the sum of the 80 scores is calculated. What are the expectation and the standard deviation of this total score?
A relay race is run between team A and team B. Each team has four runners, who successively run around a track and pass a baton to the next runner in their team when they have finished. The team finishes when the fourth runner has completed the run. The runners on team A take times with an
Suppose that a temperature has a mean of 1 10 F and a standard deviation of 2.2 F. What are the mean and the standard deviation in degrees Centigrade? (F = 9C/5 + 32)
Recall that for any function g(X) of a random variable X,where f(x) is the probability density function of X. Use this result to show that E(aX + b) = aE(X) + b and Var(a X + b) = a2Var(X)
A person's cholesterol level C can be measured by three different tests. Test-a returns a value Xa with a mean C and a standard deviation 1.2, test-β returns a value Xβ with a mean C and a standard deviation 2.4. and test-β returns a value X., with a mean C and a standard deviation 3.1. Suppose
Suppose that the impurity levels of water samples taken from a particular source are independent with a mean value of 3.87 and a standard deviation of 0.18. (a) What are the mean and the standard deviation of the sum of the impurity levels from two water samples? (b) What are the mean and the
Suppose that X1, X2, and X3 are independent random variables each with a mean of μ and a variance of σ2. Compare the means and variances of Y = 3X1 and Z = X, + X2 + X3
A machine part is assembled by fastening two components of type A and one component of type B end to end. Suppose that the lengths of components of type have an expectation of 37.0 mm and a standard deviation of 0.7 mm. whereas the lengths of components of type B have an expectation of 24.0 mm and
In a game a player either wins $10 with a probability of 1/8 or loses SI with a probability of 7/8. What are the expectation and variance of the total net winnings of a player after 50 turns at the game?
Ten cards are drawn with replacement from a pack of cards. What arc the expectation and variance of the number of Aces drawn? If the drawings are made without replacement, what is the expected number of Aces drawn?
Suppose that the random variable X has a probability density function f(x) = 2x for 0 < x < 1. Find the probability density function and the expectation of the random variable Y in the following cases. (a) Y = X3 (b) Y = √X (c) Y = 1/(1 +X) (d) K = 2X
The radius of a soap bubble has a probability density function f(r) = A(1 - (r - l)2) for 0 < r < 2. (a) What is the value of A? (b) What is the probability density function of the volume of the soap bubble? (c) What is the expected value of the volume of the soap bubble?
A box contains four red balls and two blue balls. Balls are drawn at random without replacement, and the random variable X measures the total number of balls selected up to the point when both blue balls have been selected. (a) What is the probability mass function of X? (b) What is the expected
If the random variables X and Y are related through the expression Y = aX + b show that they have a correlation of I if a > 0 and of -1 if a < 0.
An insurance company charges a customer an annual premium of $100, and there is a probability of 0.9 that the customer will not need to make a claim. If the customer does make a claim, the amount of the claim $X has a probability density functionfor 0 ‰¤ x ‰¤ 1800. Each
Suppose that telephone calls on a particular line have an expected length of 320 seconds with a standard deviation of 63 seconds, and suppose that the call lengths are independent of each other. (a) What are the expectation and the standard deviation of the total length of five calls? (b) What are
In an evaluation procedure, n items are ranked in order of effectiveness. No ties are allowed, so that the ranks are the positive integers from 1 to n. Suppose that all of the possible rankings are equally likely. (a) What is the probability mass function of the rank of a particular item? (b) What
Nancy and Tom have to take a bus ride. The time taken by a bus for the specified journey has an expectation of 87 minutes with a standard deviation of 3 minutes, and the times taken by different buses are independent of each other. Consider the random variable X, which is defined to be the sum of
When a fair coin is tossed, 10 points are scored if a head is obtained and 20 points are scored if a tail is obtained. Suppose that the coin is tossed two times, and let the random variable X be the total score obtained. (a) What is the state space of X? (b) What is the probability mass function of
A continuous random variable has a probability density function f(x) = Ax for 5 ≤ x ≤ 6. (a) What is the value of A? (b) What is the cumulative distribution function of the random variable? (c) What is the expectation of the random variable? (d) What is the standard deviation of the random
Components have a weight with an expectation of 438 and a standard deviation of 4. (a) What are the expectation and the standard deviation of the sum of the weights of three randomly selected components? (b) What are the expectation and the standard deviation of the average of the weights of eight
In a game a player rolls a fair six-sided die. If the score is even, the player receives an amount of dollars equal to the score. If the score is odd, the player receives an amount of dollars equal to the score multiplied by three. It costs the player $5 to play the game. (a) What is the
Are the following statements true or false? (a) The variance of a random variable is measured in the same units as the random variable. (b) In a diving competition, the scores awarded by judges for a particular type of dive have an expected value of 78 with a standard deviation of 5. If the scores
The probability mass function of the number of calls taken by a switchboard within 1 minute is given in Figure 2.77.(a) Compute and sketch the cumulative distribution function of the number of calls taken within a minute. (b) What is the expected number of calls taken within a minute? (c) What is
Suppose that the time taken to download a file of a certain kind onto a computer has an expected value of 22 minutes and a standard deviation of 1.8 minutes. If a technician needs to download five of these files onto a computer one after the other, and the downloading times are independent of each
When a computer chip is examined to discover how many of the solder joints have become cracked, there is a probability of 0.12 that none of the joints are cracked, a probability of 0.43 that one of the joints is cracked, a probability of 0.28 that two of the joints are cracked, and a probability of
A discrete random variable takes the values -22, 3, 19, and 23 with probabilities 0.3, 0.2, 0.1, and 0.4 respectively. (a) What is the expectation of the random variable? (b) What is the standard deviation of the random variable?
A random variable has a probability density function f(x) = A/x2 for 2 ≤ x ≤ 4. (a) What is the value of A? (b) What is the lower quartile of the distribution?
Bricks have weights that have a mean 250 and a standard deviation 4. (a) Suppose X is the weight of a randomly chosen brick. Let Y = c + dX. What are the values of C and d such that Y has a mean 100 and a standard deviation 1. (b) Suppose 10 bricks are chosen at random. What are the mean and the
An evaluation score X1 of a candidate using method 1 has a mean of 100 and a standard deviation of 12, while an evaluation score X2 of a candidate using method 2 has a mean of 100 and a standard deviation of 13. If the two evaluation scores are independent, what values of c1 and c2 can be chosen so
Wafers of type A have thicknesses with a mean of 134.9 and a standard deviation of 0.7, while wafers of type B have thicknesses with a mean of 138.2 and a standard deviation of 1.1. The thicknesses of the wafers are all independent of each other. (a) Three wafers of type A are stacked on top of
A box initially contains two red balls and two blue balls. At each turn, one of the balls within the box is selected at random and then replaced together with another ball of the opposite color. (a) After three turns, what is the distribution of the number of red balls in the box? (b) After three
Suppose that an Ace, a King, a Queen, and a Jack are all worth 15 points, and that other cards are all worth their face value. If a hand of 13 cards is dealt, what is the expected value of the total score of the hand?
The acidity level X of a soil sample has a probability density function f(x) = A(3/2)x for 1 ≤ x ≤ 11. (a) What is the value of A? (b) Compute and sketch the cumulative distribution function of X. (c) What is the median soil acidity level? (d) What is the interquartile range of the soil acidity
Suppose that the random variables X and Y have a joint probability density function f(x, y) = 4x(2 - y) for 0 ≤ x ≤ 1 and 1 ≤ y ≤ 2. (a) What is the marginal probability density function of X? (b) Are the random variables X and Y independent? (c) What is Cov(X, Y)? (d) What is the
The density X of a chemical solution is f(x) = A(x + 2/x) for 5 ≤ x ≤ 10. (a) What is the value of A? (b) Compute and sketch the cumulative distribution function of the density. (c) What is the mean density level? (d) What is the variance of the density level? (e) What is the median density
Recall that Var(aX + b) = a2Var(X) and Var(X1 + X2) = Var(X1) + Var(X1) if the random variables X1 and X2 are independent. Use these results to show that Var(a1X1 + ... + anXn + b) = a12Var(X1) + ... + an2Var(Xn) if the random variables Xi are independent.
Suppose that X ~ 5(10, 0.12). Calculate:(a) P(X = 3)(b) P(X = 6)(c) P(X < 2)(d) P(X > 7)(e) E(X) (f) Var(X)
Suppose that X ~ B(7, 0.8). Calculate:(a) P(X = 4) (b) P(X ≠ 2)(c) P(X < 3)(d) P(X > 6)(e) E(X) (f) Var(X)
Draw line graphs of the probability mass functions of a B(6. 0.5) distribution and a B(6, 0.7) distribution. Mark the expected values of the distributions on the line graphs and calculate the standard deviations of the two distributions.
An archer hits a bull's-eye with a probability of 0.09. and the results of different attempts can be taken to be independent of each other, [f the archer shoots nine arrows, calculate the probability that: (a) Exactly two arrows score bull's-eyes. (b) At least two arrows score bull's-eyes. What is
A fair die is rolled eight times. Calculate the probability that there are: (a) Exactly five even numbers (b) Exactly one 6 (c) No 4s (d) At least six prime numbers
A multiple-choice quiz consists often 1questions, each with five possible answers of which only one is correct. A student passes the quiz if seven or more correct answers are obtained. What is the probability that a student who guesses blindly at all of the questions will pass the quiz? What is the
A flu virus hits a company employing 180 people. Independent of the other employees, there is a probability of p = 0.35 that each person needs to take sick leave. What are the expectation and variance of the proportion of the workforce who need to take sick leave? In general, what value of the sick
A company receives 60% of its orders over the Internet. Within a collection of 18 independently placed orders, what is the probability that (a) between 8 and 10 of the orders are received over the Internet. (b) no more than 4 of the orders are received over the Internet? (This problem is continued
If X has a geometric distribution with parameter p = 0.7. calculate: (a) P(X = 4) (b) P(X = 1) (c) P(X < 5) (d) P(X > 8)
Consider a fair six-sided die. The die is rolled until a 6 is obtained for the third time. What is the expectation of the number of die rolls needed?
A fair coin is tossed until the fifth head is obtained. What is the probability that the coin is tossed exactly 10 times?
If X has a negative binomial distribution with parameters p = 0.6 and r = 3. calculate:(a) P(X = 5)(b) P(X = 8)(c) P(X < 7)(d) P(X > 7)
Suppose that X1.........Xr, are independent random variables, each with a geometric distribution with parameter p. Explain why Y = X1 +∙∙∙∙∙∙∙∙∙ + Xr has a negative binomial distribution with parameters p and r. Use this relationship to establish the mean and variance of a
Recall Problem 3.1.4 where an archer hits a bull's-eye with a probability of 0.09. and the results of different attempts can be taken to be independent of each other, (a) If the archer shoots a series of arrow s, w hat is the probability that the first bull's-eye is scored with the fourth
A supply container dropped from an aircraft by parachute hits a target with a probability of 0.37.(a) What is the expected number of container drops needed to hit a target?(b) If hits from three containers are required to provide? sufficient supplies, what is the expected number of "^V containers
Cards are chosen randomly from a pack of cards with replacement. Calculate the probability that:(a) The first heart is obtained on the third drawing.(b) The fourth heart is obtained on the tenth drawing. What is the expected number of cards drawn before the fourth heart is obtained? If the first
When a fisherman catches a fish, it is a young one with a probability of 0.23 and it is returned to the water. On the other hand, an adult fish is kept to be eaten later.(a) What is the expected number of fish caught by the fisherman before an adult fish is caught?(b) What is the probability that
Recall Problem 3.1.9. in which a company receives 60% of its orders over the Internet. Within a certain period of time:(a) What is the probability that the fifth order received is the first Internet order?(b) What is the probability that the eighth order received is the fourth Internet order.'
Let X have a hypergeometric distribution with N = 11. r = 6, and n = 7. Calculate: a. P(X = 4) b. P(X = 5) c. P(X < 3)
In a ground water contamination study, the researchers identify 25 possible sites for drilling and sample collection. Unknown to the researchers. 19 of these sites have ground water with a high contamination, while the other 6 sites have ground water with a low contamination. The researchers have a
A committee consists of eight right-wing members and seven left-wing members. A subcommittee is formed by randomly choosing five of the committee members. Draw a line graph ol the probability mass function of the number of right-wing members serving on the subcommittee.
A box contains 17 balls of which 10 are red and 7 are blue. A sample of 5 balls is chosen at random and placed in jar. Calculate the probability that: The jar contains exactly 3 red balls. The jar contains exactly 1 red ball. The jar contains more blue balls than red balls
A jury of 12 people is selected at random from a group of 16 men and 18 women. What is the probability that the jury contains exactly 7 women? Suppose that the jury is selected at random from a group of 1600 men and 1800 women. Use the binomial approximation to the hypergeometric distribution to
Five cards are selected at random from a pack of cards without replacement. What is the probability that exactly three of them are picture cards (Kings. Queens, or Jacks)? If a hand of 13 cards is dealt from the pack, what are the expectation and variance of the number of picture cards that it
There are 11 items of a product on a shelf in a retail outlet, and unknown to the customers. 4 of the items are outdated. Suppose that a customer takes 3 items at random. a. What is the probability that none of the outdated products are selected by the customer? b. What is the probability that
A plate has 15 cakes on it, of which 9 are chocolate and 6 are strawberry. A child randomly selects 5 of the cakes and eats them. What is the probability that the number of chocolate cakes remaining on the plate is between 5 and 7 inclusive?
(a) A box contains 8 red balls and 8 blue balls, and 4 balls arc taken at random without replacement. What is the probability that 2 red balls and 2 blue balls are taken? (b) A box contains 50,000 red balls and 50,000 blue balls, and 4 balls arc taken at random without replacement. Estimate the
If X ~ P(3.2), calculate: (a)P(X = 1) (b)P(X < 3) (c)P(X > 6) (d) P(X = 0|X < 3)
If X ~ P(2.1), calculate: (a) P(X = 0) (b) P(X < 2) (c) P(X > 5) (d) P(X =1|X < 2)
The number of cracks in a ceramic tile has a Poisson distribution with a mean of λ = 2.4. What is the probability that a tile has no cracks? What is the probability that a tile has four or more cracks?
On average there are about 25 imperfections in 100 meters of optical cable. Use the Poisson distribution to estimate the probability that there are no imperfections in 1 meter of cable. What is the probability that there is no more than one imperfection in 1 meter of cable?
On average there are four traffic accidents in a city during one hour of rush-hour traffic. Use the Poisson distribution to calculate the probability that in one such hour there arc (a) No accidents (b) At least six accidents
Recall that the Poisson distribution with a parameter value of λ = up can be used to approximate the B(n, p) distribution when n is very large and the success probability p is very small. A box contains 500 electrical switches, each one of which has a probability of 0.005 of being defective. Use
In a scanning process, the number of misrecorded pieces information has a Poisson distribution with parameter λ = 9.2. (a) What is the probability that there are between 6 and misrecorded pieces of information'.' (b) What is the probability that there are no more than 4 misrecorded pieces of
A garage sells three types of tires, type A, type B. and type C. A customer purchases type A with probability 0.23, type B with probability 0.48. and type C with probability 0.29. a. What is the probability that the next 11 customers purchase four sets of type A, five sets of type B. and two sets
A fair die is rolled 15 times. Calculate the probability that there are: a. Exactly three 6s and three 5s b. Exactly three 6s, three 5s, and four 4s c. Exactly two 6s What is the expected number of 6s obtained.
Recall Problems 3.1.4 and 3.2.5, where an archer hits a bull's-eye with a probability of 0.09. Suppose also that the archer misses the target completely with a probability of 0.12. If the areher shoots eight arrows whose performances are independent of each other, calculate the probability of: (a)
A researcher plants 22 seedlings. After one month, independent of the other seedlings, each seedling has a probability of 0.08 of being dead, a probability of 0.19 of exhibiting slow growth, a probability of 0.42 of exhibiting medium growth, and a probability of 0.31 of exhibiting strong growth.
Recall Problems 3.1.9 and 3.2.9. where a company receives 60% of its orders over the Internet. Suppose that 30% of the orders received over the Internet are large orders, and 40% of the orders received by other means are large orders. Out of eight independently placed orders, what is the
An integrated circuit manufacturer produces wafers that contain 18 chips. Each chip has a probability of 0.085 of not being placed quite correctly on the wafer. (a) Find the probability that a wafer contains at least three incorrectly placed chips. (b) What is the probability that a wafer contains
(a) If a fair die is rolled 22 times, what is the probability that a 6 is obtained exactly 3 times?(b) If a fair die is rolled, what is the probability that the third time that a 6 is obtained is on the tenth roll?(c) If a fair coin is tossed 11 times, what is the probability that three or fewer
Are the following statements true or false? (a) An unfair coin, for which the probabilities of a head and a tail are different, is tossed seven times. The probability of getting three heads and four tails cannot be the same as the probability of getting four heads and three tails. (b) A fair die is
(a) A fair die is rolled 10 times. What is the probability of obtaining the outcome 6 exactly 3 times?(b) A fair die is repeatedly rolled. What is the probability that the outcome 6 is obtained for the fourth tune on the twentieth roll?(c) A fair die is rolled nine times. What is the probability of
The number of imperfections in an object has a Poisson distribution with a mean X = 8.3. If the number of imperfections is 4 or less, the object is called "top quality." If the number of imperfections is between 5 and 8 inclusive, the object is called "good quality." If the number of imperfections
A biologist has a culture consisting of 13 cells. In a period of one hour, independent of the other cells, there is a probability of 0.4 that each of these cells splits into 2 cells. What is the probability that after one hour the biologist has at least 16 cells? What is the expected number of
A beverage company has three different formulas for its soft drink product. Tests reveal that 40% of consumers prefer formula I. 25% of consumers prefer formula II. and 35% of consumers prefer formula III. If eight consumers are chosen at random, calculate the probability that: (a) Exactly two
A company's toll-free complaints line receives an average of about 40 calls per hour. Use the Poisson distribution to estimate the probability that in one minute there are (a) No calls (b) Exactly one call (c) Three or more calls
The number of radioactive particles passing through a counter in one minute has a Poisson distribution with k = 3.3. What is the probability that in one minute there are exactly two particles passing through the counter? What is the probability that in one minute there are at least six particles
In a typical sports playoff series, two teams play a sequence of games until one team, the eventual winner, has won four games. Suppose that in each game team A beats team B with a probability of 0.55. and that the results of different games are independent.(a) Explain how the negative binomial
A golf shop sells both right-handed and left-handed of clubs, and 42% of customers purchase right-handed sets whereas 5895 of customers purchase left-handed se The owner opens the shop one day and waits for customers to arrive. (a) What is the probability that the ninth set of clubs in the day is
Box A contains six red balls and five blue balls. Box B contains five red balls and six blue balls. A fair coin is tossed, and if a head is obtained, three balls are taken random from box A and placed in jar. If a tail is obtained, three balls are taken at random from box B placed in jar. If the
Suppose that a box contains 40 items of which 4 are defective. If a random sample of 5 items is chosen, w is the probability that it contains no more than 1 defective item? If a random sample of 5 items is chosen from a collection of 4,000,000 items of which 400.000 are defective, what is the
Suppose that X ~ (7(-3, 8). Find: (a) E.(V) (b) The standard deviation of X (c) The upper quartile of the distribution (d) P(0 < X < 4)

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