 3.1E: A concrete beam may fail either by shear (S) or flexure (F). Suppos...
 3.3.1: Classify the following random variables as discrete or continuous: ...
 3.2E: Give three examples of Bernoulli rv’s (other than those in the text).
 3.3.2: An overseas shipment of 5 foreign automobiles contains 2 that have ...
 3.3E: Using the experiment in Example 3.3, define two more random variabl...
 3.3.3: Let W be a random variable giving the number of heads minus the num...
 3.4E: Let X = the number of nonzero digits in a randomly selected zip cod...
 3.3.4: A coin is ipped until 3 heads in succession occur. List only those ...
 3.5E: If the sample space will have an infinite set of possible values? I...
 3.3.5: Determine the value c so that each of the following functions can s...
 3.6E: Starting at a fixed time, each car entering an intersection is obse...
 3.3.6: The shelf life, in days, for bottles of a certain prescribed medici...
 3.7E: For each random variable defined here, describe the set of possible...
 3.3.7: The total number of hours, measured in units of 100 hours, that a f...
 3.8E: Each time a component is tested, the trial is a success (S) or fail...
 3.3.8: Find the probability distribution of the random variable W in Exerc...
 3.9E: An individual named Claudius is located at the point 0 in the accom...
 3.3.9: The proportion of people who respond to a certain mailorder solici...
 3.10E: The number of pumps in use at both a sixpump station and a fourpu...
 3.3.10: Find a formula for the probability distribution of the random varia...
 3.11E: ?Let X be the number of students who show up for a professor's offi...
 3.3.11: A shipment of 7 television sets contains 2 defective sets. A hotel ...
 3.12E: Airlines sometimes overbook flights. Suppose that for a plane with ...
 3.3.12: An investment rm oers its customers municipal bonds that mature aft...
 3.13E: A mailorder computer business has six telephone lines. Let ?X ?den...
 3.3.13: The probability distribution of X, the number of imperfections per ...
 3.14E: A contractor is required by a county planning department to submit ...
 3.3.14: The waiting time, in hours, between successive speeders spotted by ...
 3.15E: Many manufacturers have quality control programs that include inspe...
 3.3.15: Find the cumulative distribution function of the random variable X ...
 3.16E: Some parts of California are particularly earthquakeprone. Suppose...
 3.3.16: Construct a graph of the cumulative distribution function of Exerci...
 3.17E: A new battery’s voltage may be acceptable (A) or unacceptable (U). ...
 3.3.17: A continuous random variable X that can assume values between x = 1...
 3.18E: Two fair sixsided dice are tossed independently. Let M = the maxim...
 3.3.18: A continuous random variable X that can assume values between x = 2...
 3.19E: A library subscribes to two different weekly news magazines, each o...
 3.3.19: For the density function of Exercise 3.17, nd F(x). Use it to evalu...
 3.20E: ?Three couples and two single individuals have been invited to an i...
 3.3.20: For the density function of Exercise 3.18, nd F(x), and use it to e...
 3.21E: Suppose that you read through this year’s issues of the New York Ti...
 3.3.21: Consider the density function f(x)=kx, 0 <x<1, 0, elsewhere. (a) Ev...
 3.22E: Refer to Exercise 13, and calculate and graph the cdf F(x). Then us...
 3.3.22: Three cards are drawn in succession from a deck without replacement...
 3.23E: 23E
 3.3.23: Find the cumulative distribution function of the random variable W ...
 3.24E: An insurance company offers its policyholders a number of different...
 3.3.24: Find the probability distribution for the number of jazz CDs when 4...
 3.26E: Alvie Singer lives at 0 in the accompanying diagram and has four fr...
 3.3.25: From a box containing 4 dimes and 2 nickels, 3 coins are selected a...
 3.3.26: From a box containing 4 black balls and 2 green balls, 3 balls are ...
 3.27E: After all students have left the classroom, a statistics professor ...
 3.3.27: The time to failure in hours of an important piece of electronic eq...
 3.28E: Show that the cdf F(x) is a nondecreasing function; that is x1 < x2...
 3.3.28: A cereal manufacturer is aware that the weight of the product in th...
 3.29E: The pmf of the amount of memory X (GB) in a purchased flash drive w...
 3.3.29: An important factor in solid missile fuel is the particle size dist...
 3.30E: An individual who has automobile insurance from a certain company i...
 3.3.30: Measurements of scientic systems are always subject to variation, s...
 3.31E: Refer to Exercise 12 and calculate V(Y) and sY. Then determine the ...
 3.3.31: Based on extensive testing, it is determined by the manufacturer of...
 3.32E: 32E
 3.3.32: The proportion of the budget for a certain type of industrial compa...
 3.33E: Let X be a Bernoulli rv with pmf as in Example 3.18. a.? ?Compute E...
 3.3.33: Suppose a certain type of small data processing rm is so specialize...
 3.34E: Suppose that the number of plants of a particular type found in a r...
 3.3.34: Magnetron tubes are produced on an automated assembly line. A sampl...
 3.35E: A small market orders copies of a certain magazine for its magazine...
 3.3.35: Suppose it is known from large amounts of historical data that X, t...
 3.36E: Let X be the damage incurred (in $) in a certain type of accident d...
 3.3.36: On a laboratory assignment, if the equipment is working, the densit...
 3.37E: The n candidates for a job have been ranked 1, 2, 3, . . . , n. Let...
 3.3.37: Determine the values of c so that the following functions represent...
 3.38E: 38E
 3.3.38: If the joint probability distribution of X and Y is given byf(x,y)=...
 3.39E: A chemical supply company currently has in stock 100 lb of a certai...
 3.3.39: From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas,...
 3.40E: a. ?Draw a line graph of the pmf of ?X ?in Exercise 35. Then determ...
 3.3.40: A fastfood restaurant operates both a drivethrough facility and a ...
 3.41E: Use the definition in Expression (3.13) to prove that V(aX + b) = ?...
 3.3.41: A candy company distributes boxes of chocolates with a mixture of c...
 3.42E: Suppose E(X) = 5 E[X(X  1)] = 27.5 What is a.? ?E(X2)? [Hint: E[X(...
 3.3.42: Let X and Y denote the lengths of life, in years, of two components...
 3.43E: Write a general rule for E(X  c) where c is a constant. What happe...
 3.3.43: Let X denote the reaction time, in seconds, to a certain stimulus a...
 3.44E: A result called ?Chebyshev’s inequality ?states that for any probab...
 3.3.44: Each rear tire on an experimental airplane is supposed to be lled t...
 3.45E: If a ? X ? b, show that a ? E(X) ? b.
 3.3.45: Let X denote the diameter of an armored electric cable and Y denote...
 3.46E: Compute the following binomial probabilities directly from the form...
 3.3.46: Referring to Exercise 3.38, nd (a) the marginal distribution of X; ...
 3.47E: The article ?"Should You Report That FenderBender?" (Consumer Repo...
 3.3.47: The amount of kerosene, in thousands of liters, in a tank at the be...
 3.48E: 48E
 3.3.48: Referring to Exercise 3.39, nd (a) f(y2) for all values of y; (b) ...
 3.49E: ?A company that produces fine crystal knows from experience that 10...
 3.3.49: Let X denote the number of times a certain numerical control machin...
 3.50E: A particular telephone number is used to receive both voice calls a...
 3.3.50: Suppose that X and Y have the following joint probability distribut...
 3.51E: ?Three cards are drawn without replacement from the 12 face cards (...
 3.3.51: Three cards are drawn without replacement from the 12 face cards (j...
 3.52E: Suppose that 30% of all students who have to buy a text for a parti...
 3.3.52: A coin is tossed twice. Let Z denote the number of heads on the rst...
 3.53E: Exercise 30 (Section 3.3) gave the pmf of ?Y, ?the number of traffi...
 3.3.53: Given the joint density function f(x,y)=6xy 8 , 0 <x<2, 2 <y<4, 0, ...
 3.54E: ?A particular type of tennis racket comes in a midsize version and ...
 3.3.54: Determine whether the two random variables of Exercise 3.49 are dep...
 3.55E: Twenty percent of all telephones of a certain type are submitted fo...
 3.3.55: Determine whether the two random variables of Exercise 3.50 are dep...
 3.56E: The College Board reports that 2% of the 2 million high school stud...
 3.3.56: The joint density function of the random variables X and Y is f(x,y...
 3.57E: A certain type of flashlight requires two typeD batteries, and the...
 3.3.57: Let X, Y , and Z have the joint probability density function f(x,y,...
 3.58E: A very large batch of components has arrived at a distributor. The ...
 3.3.58: Determine whether the two random variables of Exercise 3.43 are dep...
 3.59E: An ordinance requiring that a smoke detector be installed in all pr...
 3.3.59: Determine whether the two random variables of Exercise 3.44 are dep...
 3.60E: A toll bridge charges $1.00 for passenger cars and $2.50 for other ...
 3.3.60: The joint probability density function of the random variables X, Y...
 3.61E: A student who is trying to write a paper for a course has a choice ...
 3.3.61: A tobacco company produces blends of tobacco, with each blend conta...
 3.62E: a. ?For fixed n, are there values of p (0 ? p ? 1) for which V(X) =...
 3.3.62: An insurance company oers its policyholders a number of dierent pre...
 3.63E: a.? ?Show that b(x; n, 1  p) = b(n  x; n, p). b. ?Show that B(x; ...
 3.3.63: Two electronic components of a missile system work in harmony for t...
 3.64E: Show that E(X) = np when X is a binomial random variable. [Hint: Fi...
 3.3.64: A service facility operates with two service lines. On a randomly s...
 3.65E: Customers at a gas station pay with a credit card (A), debit card (...
 3.3.65: Let the number of phone calls received by a switchboard during a 5...
 3.66E: An airport limousine can accommodate up to four passengers on any o...
 3.3.66: Consider the random variables X and Y with joint density function f...
 3.67E: Refer to Chebyshev’s inequality given in Exercise 44. Calculate P( ...
 3.3.67: An industrial process manufactures items that can be classied as ei...
 3.68E: Eighteen individuals are scheduled to take a driving test at a part...
 3.3.68: Consider the following joint probability density function of the ra...
 3.69E: Each of 12 refrigerators of a certain type has been returned to a d...
 3.3.69: The life span in hours of an electrical component is a random varia...
 3.70E: An instructor who taught two sections of engineering statistics las...
 3.3.70: Pairs of pants are being produced by a particular outlet facility. ...
 3.71E: A geologist has collected 10 specimens of basaltic rock and 10 spec...
 3.3.71: The shelf life of a product is a random variable that is related to...
 3.72E: A personnel director interviewing 11 senior engineers for four job ...
 3.3.72: Passenger congestion is a service problem in airports. Trains are i...
 3.73E: Twenty pairs of individuals playing in a bridge tournament have bee...
 3.3.73: Impurities in a batch of nal product of a chemical process often re...
 3.74E: A secondstage smog alert has been called in a certain area of Los ...
 3.3.74: The time Z in minutes between calls to an electrical supply system ...
 3.75E: The probability that a randomly selected box of a certain type of c...
 3.3.75: A chemical system that results from a chemical reaction has two imp...
 3.76E: A family decides to have children until it has three children of th...
 3.3.76: Consider the situation of Review Exercise 3.75. But suppose the joi...
 3.77E: Three brothers and their wives decide to have children until each f...
 3.3.77: Consider the random variables X and Y that represent the number of ...
 3.78E: According to the article “Characterizing the Severity and Risk of D...
 3.3.78: The behavior of series of components plays a huge role in scientic ...
 3.79E: 79E
 3.3.79: Another type of system that is employed in engineering work is a gr...
 3.80E: Let X be the number of material anomalies occurring in a particular...
 3.3.80: Consider a system of components in which there are 5 independent co...
 3.81E: Suppose that the number of drivers who travel between a particular ...
 3.3.81: Take 5 class periods to observe the shoe color of individuals in cl...
 3.82E: Consider writing onto a computer disk and then sending it through a...
 3.83E: An article in the Los Angeles Times (Dec. 3, 1993) reports that 1 i...
 3.84E: The ?Centers for Disease Control and Prevention reported in 2012 th...
 3.85E: Suppose small aircraft arrive at a certain airport according to a P...
 3.86E: Organisms are present in ballast water discharged from a ship accor...
 3.87E: The number of requests for assistance received by a towing service ...
 3.88E: In proof testing of circuit boards, the probability that any partic...
 3.89E: The article “ReliabilityBased ServiceLife Assessment of Aging Con...
 3.90E: Let X have a Poisson distribution with parameter ?. Show that E( X ...
 3.91E: Suppose that trees are distributed in a forest according to a twod...
 3.92E: Automobiles arrive at a vehicle equipment inspection station accord...
 3.94E: Consider a deck consisting of seven cards, marked 1, 2, . . . , 7. ...
 3.95E: After shuffling a deck of 52 cards, a dealer deals out 5. Let X = t...
 3.96E: The negative binomial rv X was defined as the number of F’s precedi...
 3.97E: Of all customers purchasing automatic garagedoor openers, 75% purc...
 3.98E: 98E
 3.99E: A koutofn system is one that will function if and only if at lea...
 3.100E: A manufacturer of integrated circuit chips wishes to control the qu...
 3.101E: Of the people passing through an airport metal detector, .5% activa...
 3.102E: An educational consulting firm is trying to decide whether high sch...
 3.103E: Consider a disease whose presence can be identified by carrying out...
 3.104E: Let p1 denote the probability that any particular code symbol is er...
 3.105E: The purchaser of a powergenerating unit requires c consecutive suc...
 3.106E: A plan for an executive travelers’ club has been developed by an ai...
 3.107E: Forty percent of seeds from maize (modernday corn) ears carry sing...
 3.108E: A trial has just resulted in a hung jury because eight members of t...
 3.109E: A reservation service employs five information operators who receiv...
 3.110E: Grasshoppers are distributed at random in a large field according t...
 3.111E: A newsstand has ordered five copies of a certain issue of a photogr...
 3.112E: Individuals A and B begin to play a sequence of chess games. Let S ...
 3.113E: A test for the presence of a certain disease has probability .20 of...
 3.114E: The generalized negative binomial pmf is given by Let X, the number...
 3.115E: There are two Certified Public Accountants in a particular office w...
 3.116E: The mode of a discrete random variable X with pmf p(x) is that valu...
 3.117E: A computer disk storage device has ten concentric tracks, numbered ...
 3.118E: If X is a hypergeometric rv, show directly from the definition that...
 3.119E: Use the fact that to prove Chebyshev’s inequality given in Exercise...
 3.120E: The simple Poisson process of Section 3.6 is characterized by a con...
 3.121E: Consider a collection A1, . . . , Ak of mutually exclusive and exha...
 3.122E: Consider a communication source that transmits packets containing d...
Solutions for Chapter 3: Mathematical Expectation
Full solutions for Probability and Statistics for Engineers and the Scientists  9th Edition
ISBN: 9780321629111
Solutions for Chapter 3: Mathematical Expectation
Get Full SolutionsSummary of Chapter 3: Mathematical Expectation
This method of relative frequencies is used to calculate the average number of heads per toss of two coins that we might expect in the long run.
Chapter 3: Mathematical Expectation includes 201 full stepbystep solutions. This textbook survival guide was created for the textbook: Probability and Statistics for Engineers and the Scientists, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Probability and Statistics for Engineers and the Scientists was written by and is associated to the ISBN: 9780321629111. Since 201 problems in chapter 3: Mathematical Expectation have been answered, more than 448037 students have viewed full stepbystep solutions from this chapter.

Additivity property of x 2
If two independent random variables X1 and X2 are distributed as chisquare with v1 and v2 degrees of freedom, respectively, Y = + X X 1 2 is a chisquare random variable with u = + v v 1 2 degrees of freedom. This generalizes to any number of independent chisquare random variables.

Alias
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.

All possible (subsets) regressions
A method of variable selection in regression that examines all possible subsets of the candidate regressor variables. Eficient computer algorithms have been developed for implementing all possible regressions

Axioms of probability
A set of rules that probabilities deined on a sample space must follow. See Probability

Backward elimination
A method of variable selection in regression that begins with all of the candidate regressor variables in the model and eliminates the insigniicant regressors one at a time until only signiicant regressors remain

Bias
An effect that systematically distorts a statistical result or estimate, preventing it from representing the true quantity of interest.

C chart
An attribute control chart that plots the total number of defects per unit in a subgroup. Similar to a defectsperunit or U chart.

Completely randomized design (or experiment)
A type of experimental design in which the treatments or design factors are assigned to the experimental units in a random manner. In designed experiments, a completely randomized design results from running all of the treatment combinations in random order.

Continuous uniform random variable
A continuous random variable with range of a inite interval and a constant probability density function.

Contour plot
A twodimensional graphic used for a bivariate probability density function that displays curves for which the probability density function is constant.

Correlation coeficient
A dimensionless measure of the linear association between two variables, usually lying in the interval from ?1 to +1, with zero indicating the absence of correlation (but not necessarily the independence of the two variables).

Counting techniques
Formulas used to determine the number of elements in sample spaces and events.

Covariance
A measure of association between two random variables obtained as the expected value of the product of the two random variables around their means; that is, Cov(X Y, ) [( )( )] =? ? E X Y ? ? X Y .

Decision interval
A parameter in a tabular CUSUM algorithm that is determined from a tradeoff between false alarms and the detection of assignable causes.

Dependent variable
The response variable in regression or a designed experiment.

Error sum of squares
In analysis of variance, this is the portion of total variability that is due to the random component in the data. It is usually based on replication of observations at certain treatment combinations in the experiment. It is sometimes called the residual sum of squares, although this is really a better term to use only when the sum of squares is based on the remnants of a modelitting process and not on replication.

Estimator (or point estimator)
A procedure for producing an estimate of a parameter of interest. An estimator is usually a function of only sample data values, and when these data values are available, it results in an estimate of the parameter of interest.

F distribution.
The distribution of the random variable deined as the ratio of two independent chisquare random variables, each divided by its number of degrees of freedom.

Fraction defective
In statistical quality control, that portion of a number of units or the output of a process that is defective.

Gamma function
A function used in the probability density function of a gamma random variable that can be considered to extend factorials