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Questions and Answers of
Statistics
In a study on eating habits, a particular participant averages 750 cm3 of food during per meal. a. It is extraordinarily rare for this participant to eat more than 1000 cm3 of food at once. Find a
An energetic student can manage to get 9 sled runs down a hill, on average, within a 30 minute time period. a. Find an upper bound on the probability that the student achieves 12 or more runs during
The average number of students in a class at a certain university is 31. a. Use the Markov inequality to find an upper bound on the probability that a class selected at random will have 40 or more
On a strand of lights, the expected number of lights that fail to work correctly is 20. a. What is an upper bound on the probability that 30 or more lights fail to work? b. If the standard deviation
Assume that the average age of tax accountants at a certain CPA firm is 44. a. Find a bound for the probability that a randomly chosen employee of the firm is 48 or older. b. If the standard
With the same assumptions as in Exercise 2, three classes are independently selected at random. a. Let A denote the event that all three of the classes have 40 or more students (i.e., 40 or more in
A basketball player has improved his scoring ability. During a game, he can be expected to make 12 shots. a. Give a bound on the probability that he makes at least 16 shots. b. If the number of shots
In a student's music library on his mp3 player, the expected length of a randomly chosen song is 3.2 minutes. a. Find an upper bound on the probability that a randomly chosen song is at least 5
The average amount of snow in a student's hometown during the winter months is 10 inches. a. Find a bound on the probability that the snowfall in a given winter will exceed 16 inches. b. If the
In the middle of a complicated dance routine, five dancers are located across the width of the stage. Let X1,..., X5 denote their positions across the stage. Assume that the Xj's are independent and
A class of 40 students is supposed to arrive at 8:30 AM, but in practice they each arrive at a time uniformly distributed between 8:20 AM and 8:35 AM, and their arrival times are independent.a. What
Three athletes, competing in the shot put event at a track and field competition, throw the ball (called the "shot") as far as they are able. The athletes have relatively similar strengths, so assume
When an extremely talented violin player is soloing, the position of her hand along the violin's neck seems to be Uniform on a 9-inch interval (the total length is about 13 inches from nut to bridge;
Alfredo, Barbara, and Cathy throw darts at a dartboard of radius 9 inches. Let X1, X2, X3 denote the distance of Alfredo, Barbara, and Cathy's darts (respectively) from the center of the board. Thus,
Suppose that three male students are making telephone calls to their girlfriends. Let X1, X2, X3 denote the times of their phone calls. Suppose that the Xj's are independent Exponential random
Let X1, X2, X3 be three independent random variables that are Uniformly distributed on the interval [0, 20]. Find the probability that the minimum of the three random variables is between 12 and 15.
Five people stand along a street waiting for a bus to arrive. The street is 9 yards away from a building. The distance (in yards) of each person from the street has density fX(x) = √x/18, for 0 ≤
The weight of a can of soda is uniformly distributed, between 238 and 242 grams. Find the density of:a. The minimum of the three weights among three cans of soda;b. The maximum of the three weights
Suppose that the delivery time for a pizza delivery is uniformly distributed between 15 to 20 minutes. Now suppose that 4 people from a dormitory independently order pizza from 4 different drivers
The time in which a student starts his homework each night-during a seven day week-is uniformly distributed between 6 PM and 8 PM. Let X and Y denote, respectively, the earliest and latest times that
If X is a Geometric random variable with parameter p (in other words,P(X = x) = px(x) = (1- p)x-lpFor x = 1, 2, 3,...,and pX(x) = 0 otherwise), then find the moment generating function of X.
If X is a Gamma random variable with parameters A and r, then find the moment generating function of X. (Use the result from Exercise 43.9 as an aid. Again, for this problem, that we are using t in
Now use the moment generating function from Exercise 43.13 to verify that, if X is Gamma with parameters λ, r, then E(X2) = r(r + 1)/λ2.
Use the moment generating function from Exercise 1 to verify that, if X is Geometric with parameter p, then E(X) = 1/p.In Exercise 1P(X = x) = px(x) = (1- p)x-lpFor x = 1, 2, 3,...,and pX(x) = 0
Now use the moment generating function from Exercise 43.1 to verify that, if X is Geometric with parameter p, then E(X2) = (2 - p)/p2. In Exercise 1 P(X = x) = px(x) = (1- p)x-lp For x = 1, 2,
If X is a Negative Binomial random variable with parameters p and r, then find the moment generating function of X. In Exercise 1P(X = x) = px(x) = (1- p)x-lpFor x = 1, 2, 3,...,and pX(x) = 0
Use the moment generating function from Exercise 5 to verify that, if X is Negative Binomial with parameters p, r, then E(X) = r/p. In Exercise 5 P(X = x) = px(x) = (1- p)x-lp For x = 1, 2,
Now use the moment generating function from Exercise 5 to verify that, if X is Negative Binomial with parameters p, r, then E(X2) = r(r + 1 - p)/p2. In Exercise 5 P(X = x) = px(x) = (1- p)x-lp For x
If X is an Exponential random variable with parameter A, then find the moment generating function of X. (Assume, for this problem, that we are using t in the range where t < λ.)
A certain type of cylindrical bottle always has height 14 cm. During the manufacturing process however, the radius of the bottom is uniformly distributed between 2.3 cm and 2.7 cm. (Whatever radius
Generalize Example 44.4 as follows:Let X be a Uniform random variable on the interval (0,1), i.e., X is Uniformly distributed with 0 < X < 1. Let Y = Xn where n is any positive integer. Find
Consider an Exponential random variable X with parameter λ > 0. Is it always true that, if a and b are positive constants, then Y = aX + b is an Exponential random variable too? If your answer is
Consider the scenario in whichU = g(X,Y) = X2AndV = h(X, Y) = X + Y.Suppose the joint density fX,Y (x,y) of X and Y is Uniform on the square where 0 ‰¤ X,Y ‰¤ 1. In other words,fX,Y(x,y) = 1if
Returning to Exercise 44.12, we still need to know the region where U and V can occur in the U, V plane, in other words, we need to identify the region where the density fU,V(u,v) is relevant. a.
A customer at the fabric store buys fabric that is 40 inches wide, i.e., 1.11 yards wide. The length is cut by the employee at the store. When she is asked to cut 1 yard of fabric, the actual length
You want to buy a square chess board from a local artists' collective. Since each chessboard is uniquely handcrafted and you didn't bring your ruler, you are not sure of the exact dimensions. Let X
Suppose that the temperature X in summer, given in Kelvin, is normally distributed with mean 300K and standard deviation 10K. Let Y be the same temperature, given in degrees Fahrenheit. So Y = 9/5
Lucas calls his girlfriend Margaret every day on his cell phone. The time X, in hours, that they talk in a day is uniformly distributed between 0 and 2. Lucas pays $2 per hour that he is on the
If the amount of cookie dough, X, used in a cookie is uniformly distributed between 1.7 and 2.6 tablespoons, then the height Y of the cookie is Y = 3/10X1/3. Find P(Y > 0.4).
Sally owns a burger restaurant. The cost of a pound of beef is $3.75. Each cheeseburger contains an amount of beef that is uniformly distributed between 0.23 and 0.27 pounds. Sally includes a slice
Let X be a random variable that is Uniformly distributed on the interval [0, π/2]. Find the expected value of Y = sin x.
A player rolls a huge bag of 200 dice. Approximate the probability that 40 or more l's appear.
The flight time of a plane flight from Denver to New York City is normally distributed, with average flight time of 3 hours and 16 minutes, and standard deviation of 30 minutes. What is the
Joe finds that, when he waits for a bus, his waiting time is Exponential, with an average waiting time of 6 minutes. Assume that he waits for one bus in the morning and again for one bus in the
People shopping at the grocery store are interviewed to see whether they enjoy artichokes. Only 11% of people like artichokes. a. How many people does the interviewer expect to meet until finding the
There are 10,000 prison inmates in a certain state. Independently of each other, and independent of their behavior on previous days, assume that, on a given day, a prisoner has probability p =
A group of scientists is testing the learning ability of monkeys. They want to see if monkeys have any language preferences. They acquire a monkey that has lived its entire life in Moscow, and they
A company has a strange policy: It sells inexpensive boots, but it only ships one boot at a time, and it does not tell the buyer whether he will receive a left or right boot (hence, the need to sell
In a certain game, a player wants to roll five dice and get as many l's as possible. Here is the scheme: Round 1: The player rolls all five of the dice, and notices how many l's appear. Round 2: The
In a large bag of 40 Starburst candies, there are 8 orange, 9 yellow, 12 red, and 11 pink. You only like orange and red. If you take 8 from the bag, what is the probability that at least 5 out of the
Study time. Let X be the time (in hours) that Stephen spends studying on one particular day during "dead week" (the nickname for the week before final exams). Then X has density 1/4e-x/4. Let Y be
On a 50-question Geography exam, the average score is 25.5 out of 50. The standard deviation of the score is 8. Find a bound on the probability that a randomly selected student's score is greater
Consider n pairs of husbands and wives, sitting randomly in a row of 2n chairs. What is the probability that each person is sitting beside her/his spouse, and also no two women are adjacent, and no
An evil scientist has devised an experiment that takes one hour to run to completion. It has a 1/12 chance of causing the world to explode. Otherwise, it fails and the evil scientist immediately
Let X and Y be independent continuous random variables that are each Uniformly distributed in the interval [0, 30]. Let Z denote the larger of X and Y; in other words, Z = max(X, Y).a. Find the
Approximately 1 in 4 relationships begin online. Suppose that we interview couples until we find 10 couples who met online. a. What is the expected number of couples we will need to interview? b.
Consider a sequence of independent flips of a biased coin. The coin is a Head or Tail with probabilities p or q : = 1 - p, respectively. Let X be the smallest j such that the jth and (j + l)st flips
A deck is thoroughly shuffled, and 5 cards are chosen, without replacement. What is the probability that the selection of 5 cards contains at least one card from each of the 4 suits?
Let X,Y be independent exponential (1) random variables.a. Compute the conditional probability that X < In 2, given X - Y ≥ 0. In other words, findP(X < In 2 | X -Y ≥ 0).b. Compute the
There are many kinds of drinks in a large cooler in the cafeteria. Fifteen percent of them are decaffeinated. After class (five days a week), Alice always needs a drink. She is always running to her
Suppose one is dealing a hand of Texas Hold 'Em poker at a standard 9-man table (each of the 9 people receives 2 cards, and there are 5 additional community cards placed face-up in front of the
There are 50 people in a tennis club. In their professional lives, 25 of the people are accordion players, 15 are basketball players, and 10 are carrot farmers. (People tend to register for the
While doing laundry, it seems that each student spends between 3 to 15 minutes in the laundry room per week, and the distribution for each student is assumed to be Uniform on this interval.a. How
For the Reddy Mikks model, construct each of the following constraints and express it with a linear left-hand side and a constant right-hand side: (a) The daily demand for interior paint exceeds that
Determine the best feasible solution among the following (feasible and infeasible) solutions of the Reddy Mikks model:(a) XI = 1, X2 = 4.(b) Xl = 2, X2 = 2.(c) XI = 3, x2 = 1.5.(d) XI = 2, X2 = 1.(e)
For the feasible solution XI = 2, x2 = 2 of the Reddy Mikks model, determine the unused amounts of raw materials Ml and M2.
Suppose that Reddy Mikks sells its exterior paint to a single wholesaler at a quantity discount. 1l1e profit per ton is $5000 if the contractor buys no more than 2 tons daily and $4500 otherwise.
Determine the feasible space for each of the following independent constraints, given that X1, x2 ≥ 0. (a) - 3x1 + x2 ≤ 6. (b) x1 - 2x2 ≥ 5. (c) 2x1 - 3x2 ≤ 12. (d) x1 - x2 ≤ 0. (e) - x1 +
In the Ma-and-Pa grocery store, shelf space is limited and must be used effectively to increase profit. Two cereal items, Grano and Wheatie, compete for a total shelf space of 60 ft2. A box of Grano
Jack is an aspiring freshman at Diem University. He realizes that "all work and no play make Jack a dull boy." As a result, Jack wants to apportion his available time of about 10 hours a day between
Wild West produces two types of cowboy hats. A type 1 hat requires twice as much labor time as a type 2. If the all available labor time is dedicated to Type 2 alone, the company can produce a total
Show & Sell can advertise its products on local radio and television (TV). The advertising budget is limited to $10,000 a month. Each minute of radio advertising costs $15 and each minute of TY
Wyoming Electric Coop owns a steam-turbine power-generating plant. BecauseWyoming is rich in coal deposits; the plant generates its steam from coal. 111is, however, may result in emission that does
Top Toys is planning a new radio and TV advertising campaign. A radio commercial costs $300 and a TV ad costs $2000. A total budget of $20,000 is allocated to the campaign. However, to ensure that
Identify the direction of increase in z in each of the following cases: (a) Maximize z = x1 - x2 (b) Maximize z = - 5x1 - 6x2. (c) Maximize z = - x1 + 2x2. (d) Maximize z = - 3x1 + x2.
Determine the solution space and the optimum solution of the Reddy Mikks model for each of the following independent changes: (a) The maximum daily demand for exterior paint is at most 2.5 tons. (b)
A company that operates 10 hours a day manufactures two products on three sequential processes. The following table summarizes the data of the problem:Determine the optimal mix of the two products.
A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of both A and B. However, the company cannot sell more than 100 units of A per day. Both products
Alum co manufactures aluminum sheets and aluminum bars. The maximum production capacity is estimated at either 800 sheets or 600 bars per day. The maximum daily demand is 550 sheets and 580 bars. The
An individual wishes to invest $5000 over the next year in two types of investment: Investment A yields 5% and investment B yields 8%. Market research recommends an allocation of at least 25% in A
The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester. The courses offered are usually of two types: practical, such as woodworking, word
ChemLabs uses raw materials I and II to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials I and II are 150 and 145 units, respectively.One unit of solution A
Identify the direction of decrease in z in each of the following cases: (a) Minimize z = 4x1 - 2x2. (b) Minimize z = - 3x1 + x2. (c) Minimize z = - x1 - 2x2.
For the diet model, suppose that the daily availability of corn is limited to 450 lb. Identify the new solution space, and determine the new optimum solution.
For the diet model, what type of optimum solution would the model yield if the feed mix should not exceed 800 Ib a day? Does the solution make sense?
John must work at least 20 hours a week to supplement his income while attending school. He has the opportunity to work in two retail stores. In store 1, he can work between 5 and 12 hours a week,
Oil Co is building a refinery to produce four products: diesel, gasoline, lubricants, and jet fuel. The minimum demand (in bbl/day) for each of these products is 14,000,30,000, 10,000, and 8,000,
Day Trader wants to invest a sum of money that would generate an annual yield of at least $10,000. Two stock groups are available: blue chips and high tech, with average annual yields of 10% and 25%,
An industrial recycling center uses two scrap aluminum metals, A and B, to produce a special alloy. Scrap A contains 6% aluminum, 3% silicon, and 4% carbon. Scrap B has 3% aluminum, 6% silicon, and
In the Reddy Mikks model (Example 2.2-1), consider the feasible solution Xl = 3 tons and X2 = 1 ton. Determine the value of the associated slacks for raw materials M1 and M2.
In the diet model (Example 2.2-2), determine the surplus amount of feed consisting of 500 Ib of corn and 600 lb of soybean meal.
Consider the following inequality 10XI - 3X2 2 ≥ - 5 Show that multiplying both sides of the inequality by -1 and then converting the resulting inequality into an equation is the same as converting
Two different products, PI and Pl, can be manufactured by one or both of two different machines, M1 and M2. The unit processing time of either product on either machine is the same. The daily
Show how the following objective function can be presented in equation form: Minimize z = max {|x1 - x2 + 3x3|, | - x1 + 3x2 - x3|} X1, x2, x3 ≥ 0 (Hint: |a| ≤ b is equivalent to a ≤ b and a
Show that the m equations: ∑nj = 1 aijXj = bi, i = 1,2, ... , m are equivalent to the following m + 1 inequalities: ∑nj = 1 aijXj = bi, i = 1,2, ... , m ∑nj = 1 (∑mj = 1 aij) x1 ≥ ∑mj = 1
Mc Burger fast-food restaurant sells quarter-pounders and cheeseburgers. A quarter pounder uses a quarter of a pound of meat, and a cheeseburger uses only .2 lb. The restaurant starts the day with
Two products are manufactured in a machining center. The productions times per unit of products 1 and 2 are 10 and 12 minutes, respectively. The total regular machine time is 2500 minutes per day. In
Jo Shop manufactures three products whose unit profits are $2, $5, and $3, respectively.The company has budgeted 80 hours of labor time and 65 hours of machine time for the production of three
In an LP in which there are several unrestricted variables, a transformation of the type
Consider the following LP:Maximize z = 2x1 + 3x2subject toX1 + 3x2 ≤ 63x1 + 2x2 ≤ 6X1, x2 ≥ 0(a) Express the problem in equation form.(b) Determine all the basic solutions of the problem, and
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