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probability statistics

Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye - Solutions

Thinning of a Poisson distribution. Suppose that the number of eggs an insect lays is Poisson distributed with parameter . Out of each egg, a larva hatches with probability p, independently of all other eggs. Find the distribution of the number of larvae.
S Convolution of Cauchy distributions (Huygens’ principle). Consider the situation in Problem 2.5 and show that ca ? cb D caCb for a; b > 0. In other words, the distribution of light on a straight line at distance aCb from the light source is the same as if every light point X on the straight
Convolutions of gamma and negative binomial distributions. Show that(a) for ˛; r; s > 0, we have ˛;r ? ˛;s D ˛;rCs ;(b) for p 2 0; 1OE and r; s > 0 we have Br;p ? Bs;p D BrCs;p. Hint: The Pólya distribution provides you with a useful identity for negative binomial coefficients.
S Dice paradox. Two dice D1 and D2 are labelled as follows.D1 W 6 3 3 3 3 3 ; D2 W 5 5 5 2 2 2 :Andy and Beth roll D1 and D2 respectively. Whoever gets the higher number wins.(a) Show that Andy has a higher probability of winning; we write this as D1 D2.(b) Beth notices this and suggests to Andy:
Coin tossing paradox. Alice suggests the following game to Bob: ‘You randomly choose two integers X; Y 2 Z withX 0 for every k 2 Z. She guesses that the coin was showing tails if the number Bob is announcing is at least Z, otherwise she guesses heads. Set up a stochastic model and find the
S Let X; Y be i.i.d. random variables taking values in ZC. Suppose that either(a) P.X D kjXCY D n/ D 1=.nC1/ for all 0 k n, or(b) P.X D kjXCY D n/ Dn k2n for all 0 k n, provided n is such that the condition has positive probability. Find the common distribution of X and Y .
Consider a fair tetrahedral dice, whose faces are numbered by 1, 2, 3, 4, and which is thrown twice. Let X be the sum and Y the maximum of the two respective numbers on the faces falling downside.(a) Find the joint distribution P ı .X; Y /1 of X and Y .(b) Construct two random variables X0 and Y
A system consists of four components that are similar but work independently. To operate properly, it is necessary that (A and B) or (C and D) are working.Let T be the failure time of the complete system, and Tk the failure time of component k 2 ¹A;B;C;Dº. Suppose the Tk have the exponential
Let X; Y be independent random variables that are exponentially distributed with parameter˛ > 0. Find the distribution density of X=.X C Y /.
S Let X be a real-valued random variable on a probability space . ;F; P/. Show that X is independent of itself if and only if X is constant with probability 1, i.e., if there exists a constant c 2 R such that P.X D c/ D 1. Hint: Consider the distribution function of X.
In number theory, Euler’s '-function is defined as the mapping ' W N ! N such that'.1/ D 1 and '.n/ D the number of integers in n D ¹1; : : : ; nº that are relatively prime to n, if n 2. Show that if n D pk1 1 : : : pkm m is the prime factorisation of n into pairwise distinct primes p1; : : :
Let . ;F; P/ be a probability space and A;B;C 2 F. Show directly (without using Corollary (3.20) and Theorem (3.24)):(a) If A and B are independent, then so are A and Bc.(b) If A;B;C are independent, then so are A [ B and C.
Generalise Pólya’s urn model to the case when the balls can take colours from a finite set E (instead of only red and white), and find the distribution of the histogram Rn after n draws; cf. equation (2.7). Can you also generalise the previous Problem 3.4 to this case? (The corresponding
S Beta-binomial representation of the Pólya distribution. Consider Pólya’s urn model with parameters a D r=c > 0 and b D w=c > 0. Let Rn be the number of red balls obtained in n draws. Use the recursive formula (2.23) to show thatfor all 0 ` n. (Hence, the Pólya model is equivalent to
You are flying from Munich to Los Angeles and stop over in London and New York.At each airport, including Munich, your suitcase must be loaded onto the plane. During this process, it will get lost with probability p. In Los Angeles you notice that your suitcase hasn’t arrived. Find the
Prisoners’ paradox. Three prisoners Andy, Bob and Charlie are sentenced to death. By drawing lots, where each had the same chance, one of the prisoners was granted pardon. The prisoner Andy, who has a survival probability of 1=3, asks the guard, who knows the result, to tell him one of his fellow
A shop is equipped with an alarm system which in case of a burglary alerts the police with probability 0.99. During a night without a burglary, a false alarm is set off with probability 0.002 (e.g. by a mouse). The probability of a burglary on a given night is 0.0005. An alarm has just gone off.
Extending the Poincaré–Borel theorem. Prove the following stronger version of Theorem(2.24). If Xi W N ! R denotes the projection onto the i th coordinate, then lim N!1 PNXi 2 OEai; bi for 1 i kD Yk iD1 N0;vOEai; bifor all k 2 N and all ai; bi 2 R with ai < bi for 1 i k. That is,
Affine transformations of normal distributions. Let X be a real random variable with normal distribution Nm;v, and a; b 2 R, a ¤ 0. Show that the random variable aXCb has the distribution N amCb;a2v.
S Gamma and beta distributions. In the situation of Section 2.5.3, let .sn/n1 be a sequence in 0;1OE with n=sn ! ˛ > 0. Show that for all r 2 N and t > 0,˛;r .0; t/ D lim n!1 P.snTrWn t/ :Can you rephrase this result in terms of random points on the positive time axis?
Gamma and negative binomial distributions. Let r 2 N, ˛; t > 0, .pn/n1 a sequence in 0; 1OE with npn ! ˛, and .tn/n1 a sequence in ZC with tn=n ! t . Show that˛;r .0; t/ D lim n!1 Br;pn.¹0; : : : ; tnº/ ;and interpret this result in terms of waiting times. Hint: First show that Br;p.¹0; 1;
Banach’s matchbox problem. The Polish mathematician Stefan Banach (1892–1945)always carried one matchbox in each of the two pockets of his coat. With equal probability, he used the matchbox in the left or in the right pocket. On finding an empty box, he replaced both boxes by new ones. Find the
S Fixed points of a random permutation. Before a theatre performance, n people leave their coats in the cloak room. After the performance, due to a power cut, the coats are returned in the dark in random order. Let X be the random number of people who get their own coat back. Find the distribution
S Projecting the multinomial distribution. Let E be a finite set, a discrete density on E, n 2 N, and X D .Xa/a2E a random variable with values inand multinomial distributionMn;. Show that, for every a 2 E, Xa has the binomial distribution Bn;.a/. = {k = (ka)aE ZE: ka = n} a E
The genome of the fruit fly Drosophila melanogaster is divided into approximately m D 7000 sections (which can be identified by the colouring of the giant polytene chromosomes found in the salivary glands). As a simplification, suppose that each section contains the same number of M D 23000 base
The interval OE0; 2 is split in two parts by picking a point at random in OE0; 1 according to the uniform distribution. Let X D l1=l2 be the ratio of the length of the shorter part l1 to the length l2 of the longer one. Find the distribution density of X.
S Simple symmetric random walk. In the evening of an election day, the votes for two competing candidates A and B are being counted. Both candidates are equally popular, i.e., on each ballot A or B are chosen with equal probability 1=2; in total 2N ballots have been cast. Set Xi D 1 or 1 depending
In the surroundings of each of 10 nuclear power stations, 100 people (chosen ‘with replacement’)are examined for a certain disease, which on average can be found in 1% of the nation’s total population. The agreement is that a power station is considered suspicious if at least 3 out of the 100
Light intensity. A light source is placed at a distancea > 0from a straight line. It radiates uniformly in all directions that eventually hit the line. Denote by X the point where a light ray hits the straight line. Show that X has the distribution density ca.x/ D a=..a2Cx2// on R.The
Buffon’s needle problem (stated by G.-L. L. Comte de Buffon in 1733 and analysed in 1777). Think of (infinitely many) parallel straight lines at distancea, embedded in a plane.A needle of length l < a is randomly placed onto the plane. What is the probability that the needle hits one of the
S Recall the situation of Bertrand’s paradox, and let X be the distance of the random chord to the centre of the circle. Find the distribution density of X if(a) the midpoint of the chord is uniformly distributed on the disk 1,(b) the angle under which the cord is seen from the centre of the
S Consider a system of n indistinguishable particles, each of which can be located in one of N different cells. Na of the cells belong to an energy level a 2 E, E a finite set. Under the assumption of the Bose–Einstein distribution, determine the probability that, for each a 2 E, ka particles
At a tombola, the winning lot shall be drawn by a ‘good luck fairy’ born on a Sunday.How many ladies must be present so that, with a probability of at least 99%, at least one of them is a Sunday child? Set up a suitable model.
S Transformation to uniformity. Prove the following converse to Proposition (1.30): If X is a real random variable with a continuous distribution function FX D F , then the random variable F.X/ is uniformly distributed on OE0; 1. Show further that the continuity of F is necessary for this to hold.
Consider the two cases(a) D OE0;1OE , .!/ D e!, X.!/ D .!=˛/1=ˇ for ! 2 and ˛; ˇ > 0,(b) D=2; =2OE , .!/ D 1=, X.!/ D sin2 ! for ! 2 .In each case, show that is a probability density and X a random variable on . ;B/, and calculate the distribution density of X with respect to the
Properties of distribution functions. Let P be a probability measure on .R;B/ and F.c/ D P.1; c/, for c 2 R, its distribution function. Show that F is increasing and right-continuous, and (1.29) holds.
S Let . ;F/ D .R;B/ and X W ! R be an arbitrary real function. Verify the following:(a) If X is piecewise monotone (i.e., R may be decomposed into at most countably many intervals, on each of which X is either increasing or decreasing), then X is a random variable.(b) If X is differentiable with
Let X; Y;X1;X2; : : : be real random variables on an event space . ;F/. Prove the following statements.(a) .X; Y / W ! R2 is a random variable.(b) X C Y and XY are random variables.(c) supn2N Xn and lim supn!1 Xn are random variables (taking values in NR).(d) ¹X D Y º 2 F, ¹limn!1 Xn existsº 2
S The rencontre problem. Alice and Bob agree to play the following game: From two completely new, identical sets of playing cards, one is well shuffled. Both piles are put next to each other face down, and then revealed card by card simultaneously. Bob bets (for a stake of e 10) that in this
The birthday paradox. Let pn be the probability that in a class of n children at least two have their birthday on the same day. For simplicity, we assume here that no birthday is on February 29th, and all other birthdays are equally likely. Show (using the inequality 1x ex) that pn 1
S Alice and Bob agree to play a fair game over 7 rounds. Each of them pays e5 as an initial stake, and the winner gets the total of e 10. At the score of 2 : 3 they have to stop the game. Alice suggests to split the winnings in this ratio. Should Bob accept the offer? Set up an appropriate model
In a pack of six chocolate drinks every carton is supposed to have a straw, but it is missing with probability 1=3, with probability 1=3 it is broken and only with probability 1=3 it is in perfect condition. Let A be the event ‘at least one straw is missing and at least one is in perfect
A certain Chevalier de Méré, who has become famous in the history of probability theory for his gambling problems and their solutions by Pascal, once mentioned to Pascal how surprised he was that when throwing three dice he observed the total sum of 11 more often than the sum of 12, although 11
Bonferroni inequality. Let A1; : : : ; An be any events in a probability space . ;F; P/.Show that PSn iD1 AiXn iD1 P.Ai / X 1i
S Inclusion–exclusion principle. Let . ;F; P/ be a probability space and Ai 2 F, i 2 I D ¹1; : : : ; nº. For J I let BJ D\j2J Aj \\j2InJ Ac jI by convention, an intersection over an empty index set is equal to . Show the following:(a) For all K I , P T k2K AkD XKJI P.BJ /:(b) For all J
S Let . i ;Fi /, i D 1; 2, be two event spaces and !1 2 1. Show the following. For every A 2 F1˝F2, the ‘!1-section’ A!1 WD ¹!2 2 2 W .!1; !2/ 2 Aº of A belongs to F2, and if f is a random variable on . 1 2;F1 ˝F2/ then the function f .!1; / is a random variable on . 2;F2/.
Let Ei , i 2 N, be countable sets and D Qi1 Ei their Cartesian product. Denote by Xi W ! Ei the projection onto the i th coordinate. Show that the system G D®¹X1 D x1; : : : ; Xk D xkº W k 1; xi 2 Ei¯[®¿¯is an intersection-stable generator of the product -algebra Ni1P.Ei /.
Let Rn be at most countable. Show that BnD P. /.
S Show that the Borel -algebra Bn on Rn coincides with B˝n, the n-fold product of the Borel -algebra B on R.
Let be uncountable and G D ¹¹!º W ! 2 º the system of the singleton subsets of .Show that .G / D ¹A W A or Ac is countableº:
Let . ;F/ be an event space, A1;A2; : : : 2 F and A D ¹! 2 W ! 2 An for infinitely many nº:Show that (a) A D TN1 SnN An, (b) 1A D lim supn!1 1An.
A control valve needs to be very sensitive to the input voltage, thus generating a good output voltage. An engineer turns the control bolts to change the input voltage. The book SN-Ratio for the Quality Evaluation, published by the Japanese Standards Association (1988), described a study on how
The manufacturer of a certain brand of freeze-dried coffee hopes to shorten the process time without jeopardizing the integrity of the product. The process engineer wants to use 3 temperatures for the drying chamber and 4 drying times. The current drying time is 3 hours at a temperature of
In the book SN-Ratio for the Quality Evaluation, published by the Japanese Standards Association (1988), a study on how tire air pressure affects the maneuverability of an automobile was described. Three different tire air pressures were compared on three different driving surfaces. The three air
Exercise 14.25 on page 608 describes an experiment involving the extraction of polyethylene through use of a solvent.1. Do a different sort of analysis on the data. Fit an appropriate regression model with a solvent categorical variable, a temperature term, a time term, a temperature by time
To ascertain the number of tellers needed during peak hours of operation, data were collected by an urban bank. Four tellers were studied during three “busy” times: (1) weekdays between 10:00 and 11:00 A.M., (2) weekday afternoons between 2:00 and 3:00 P.M., and (3) Saturday mornings between
In the experiment of Review Exercise 14.35, cake volume was also used as a response. The units are cubic inches. Test for interaction between factors and discuss main effects. Assume that both factors are fixed effects.
An experiment was conducted in the Department of Food Science and Technology at Virginia Tech. It was of interest to characterize the texture of certain types of fish in the herring family. The effect of sauce types used in preparing the fish was also studied. The response in the experiment was
The Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech at Virginia Tech was involved in analyzing a set of data taken by personnel in the Human Nutrition, Foods, and Exercise Department in which it was of interest to study the effects of flour type and percent sweetener on
Personnel in the Materials Science and Engineering Department at Virginia Tech conducted an experiment to study the effects of environmental factors on the stability of a certain type of copper-nickel alloy. The basic response was the fatigue life of the material. The factors are level of stress
A study was made to determine if humidity conditions have an effect on the force required to pull apart pieces of glued plastic. Three types of plastic were tested using 4 different levels of humidity. The results, in kilograms, are as follows:1. Assuming a fixed effects experiment, perform an
A process engineer wants to determine if the power setting on the machines used to fill certain types of cereal boxes results in a significant effect on the actual weight of the product. The study consists of 3 randomly chosen types of cereal manufactured by the company and 3 fixed power settings.
A manufacturer of latex house paint (brand A) would like to show that its paint is more robust to the material being painted than that of its two closest competitors. The response is the time, in years, until chipping occurs. The study involves the three brands of paint and three randomly chosen
A defense contractor is interested in studying an inspection process to detect failure or fatigue of transformer parts. Three levels of inspections are used by three randomly chosen inspectors. Five lots are used for each combination in the study. The factor levels are given in the data. The
Consider the following analysis of variance for a random effects experiment:Test for significant variance components among all main effects and interaction effects at the 0.01 level of significance 1. by using a pooled estimate of error when appropriate;2. by not pooling sums of squares of
A defense contractor is interested in studying an inspection process to detect failure or fatigue of transformer parts. Three levels of inspections are used by three randomly chosen inspectors. Five lots are used for each combination in the study. The factor levels are given in the data. The
To estimate the various components of variability in a filtration process, the percent of material lost in the mother liquor is measured for 12 experimental conditions, with 3 runs on each condition. Three filters and 4 operators are selected at random for use in the experiment.1. Test the
Assuming a random effects experiment for Exercise 14.6 on page 596, estimate the variance components for brand of orange juice concentrate, for number of days from when orange juice was blended until it was tested, and for experimental error.
Factorial Experiments for Random Effects and Mixed Models In a two-factor experiment with random effects, we have the model for i = 1, 2, …, a; j = 1, 2, …, b; and k = 1, 2, …, n, where the A , B ,(AB) , and ϵ are independent random variables with means 0 i j ij ijk and variances , and σ ,
In the book Design of Experiments for Quality Improvement, published by the Japanese Standards Association(1989), a study is reported on the extraction of polyethylene by using a solvent and how the amount of gel (proportion) is influenced by three factors: the type of solvent, extraction
A scientist collects experimental data on the radius of a propellant grain, y, as a function of powder temperature, extrusion rate, and die temperature. Results of the three-factor experiment are as follows:Resources are not available to make repeated experimental trials at the eight combinations
Consider combinations of three factors in the removal of dirt from standard loads of laundry. The first factor is the brand of the detergent, X, Y, or Z. The second factor is the type of detergent, liquid or powder. The third factor is the temperature of the water, hot or warm. The experiment was
Consider the data set in Exercise 14.21.1. Construct an interaction plot for any two-factor interaction that is significant.2. Do a normal probability plot of residuals and comment.
Electronic copiers make copies by gluing black ink on paper, using static electricity. Heating and gluing the ink on the paper comprise the final stage of the copying process. The gluing power during this final process determines the quality of the copy. It is postulated that temperature, surface
For a study of the hardness of gold dental fillings, five randomly chosen dentists were assigned combinations of three methods of condensation and two types of gold. The hardness was measured. (See Hoaglin, Mosteller, and Tukey, 1991.) Let the dentists play the role of blocks. The data are
Corrosion fatigue in metals has been defined as the simultaneous action of cyclic stress and chemical attack on a metal structure. In the study Effect of Humidity and Several Surface Coatings on the Fatigue Life of 2024-T351 Aluminum Alloy, conducted by the Department of Mechanical Engineering at
Consider an experimental situation involving factors A, B, and C, where we assume a three-way fixed effects model of the form y = μ + α + β + γ + (βγ) + ϵ . All other interactions are considered to be nonexistent or negligible. The data are presented here.1. Perform a test of significance on
The following data are measurements from an experiment conducted using three factors A, B, and C, all fixed effects:ijkl i j k jk ijkl 1. Perform tests of significance on all interactions at the α = 0.05 level.2. Perform tests of significance on the main effects at the α =0.05 level.3. Give an
The purpose of the study The Incorporation of a Chelating Agent into a Flame Retardant Finish of a Cotton Flannelette and the Evaluation of Selected Fabric Properties, conducted at Virginia Tech, was to evaluate the use of a chelating agent as part of the flame retardant finish of cotton
In Myers, Classical and Modern Regression with Applications (Duxbury Classic Series, 2nd edition, 1990), an experiment is described in which the Environmental Protection Agency seeks to determine the effect of two water treatment methods on magnesium uptake. Magnesium levels in grams per cubic
Two factors in a manufacturing process for an integrated circuit are studied in a two-factor experiment. The purpose of the experiment is to learn their effect on the resistivity of the wafer. The factors are implant dose (2 levels) and furnace position (3 levels). Experimentation is costly so only
A study was done to determine the impact of two factors, method of analysis and the laboratory doing the analysis, on the level of sulfur content in coal. Twenty-eight coal specimens were randomly assigned to 14 factor combinations, the structure of the experimental units represented by
In an experiment conducted in the Civil Engineering Department at Virginia Tech, growth of a certain type of algae in water was observed as a function of time and the dosage of copper added to the water. The data are as follows. Response is in units of algae.1. Do an analysis of variance and show
An engineer is interested in the effects of cutting speed and tool geometry on the life in hours of a machine tool. Two cutting speeds and two different geometries are used. Three experimental tests are accomplished at each of the four combinations. The data are as follows.1. Show an
The extraction rate of a certain polymer is known to depend on the reaction temperature and the amount of catalyst used. An experiment was conducted at four levels of temperature and five levels of the catalyst, and the extraction rate was recorded in the following table.Perform an analysis of
To ascertain the stability of vitamin C in reconstituted frozen orange juice concentrate stored in a refrigerator for a period of up to one week, the study Vitamin C Retention in Reconstituted Frozen Orange Juice was conducted by the Department of Human Nutrition, Food and Exercise at Virginia
To determine which muscles need to be subjected to a conditioning program in order to improve one’s performance on the flat serve used in tennis, a study was conducted by the Human Nutrition, Foods, and Exercise Department at Virginia Tech. Five different muscles were tested on each of 3
An experiment was conducted to determine whether additives increase the adhesiveness of rubber products. Sixteen products were made with the new additive and another 16 without the new additive. The observed adhesiveness was as recorded below.Perform an analysis of variance to test for significant
Three strains of rats were studied under 2 environmental conditions for their performance in a maze test. The error scores for the 48 rats were recorded.Use a 0.01 level of significance to test the hypothesis that 1. there is no difference in error scores for different environments;2. there is no
Corrosion fatigue in metals has been defined as the simultaneous action of cyclic stress and chemical attack on a metal structure. A widely used technique for minimizing corrosion fatigue damage in aluminum involves the application of a protective coating. A study conducted by the Department of
An experiment was conducted to study the effects of temperature and type of oven on the life of a particular component. Four types of ovens and 3 temperature levels were used in the experiment. Twenty-four pieces were assigned randomly, two to each combination of treatments, and the following
Group Project: It is of interest to determine which type of sports ball can be thrown the longest distance. The competition involves a tennis ball, a baseball, and a softball. Divide the class into teams of five individuals. Each team should design and conduct a separate experiment. Each team
For the randomized block design with k treatments and b blocks, show that Figure 13.16: SAS printout for Review Exercise 13.48.Figure 13.17: SAS printout for Review Exercise 13.50.
Show that the computing formula for SSB, in the analysis of variance of the randomized complete block design, is equivalent to the corresponding term in the identity of Theorem 13.3.2
Prove Theorem 13.2.
Show that the mean square error for the analysis of variance in a one-way classification is an unbiased estimate of σ .
A study is conducted to compare gas mileage for 3 competing brands of gasoline. Four different automobile models of varying size are randomly selected. The data, in miles per gallon, follow. The order of testing is random for each model.1. Discuss the need for the use of more than a single model of
An experiment was conducted to compare three types of paint for evidence of differences in their wearing qualities.They were exposed to abrasive action and the time in hours until abrasion was noticed was observed. Six specimens were used for each type of paint. The data are as follows.1. Do an
A company that stamps gaskets out of sheets of rubber, plastic, and cork wants to compare the mean number of gaskets produced per hour for the three types of material. Two randomly selected stamping machines are chosen as blocks.The data represent the number of gaskets (in thousands)produced per
In a study that was analyzed for personnel in the Department of Biochemistry at Virginia Tech, three diets were given to groups of rats in order to study the effect of each on dietary residual zinc in the bloodstream. Five pregnant rats were randomly assigned to each diet group, and each was given

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