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In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturmâ€“Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact wellposed.
Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.
Initial value problem
A more mathematical way to picture the difference between an initial value problem and a boundary value problem is that an initial value problem has all of the conditions specified at the same value of the independent variable in the equation (and that value is at the lower boundary of the domain, thus the term "initial" value). On the other hand, a boundary value problem has conditions specified at the extremes of the independent variable. For example, if the independent variable is time over the domain [0,1], an initial value problem would specify a value of y(t) and y'(t) at time t=0, while a boundary value problem would specify values for y(t) at both t=0 and t=1.
If the problem is dependent on both space and time, then instead of specifying the value of the problem at a given point for all time the data could be given at a given time for all space. For example, the temperature of an iron bar with one end kept at absolute zero and the other end at the freezing point of water would be a boundary value problem.
Concretely, an example of a boundary value (in one spatial dimension) is the problem
 y(x)+y(x)=0 \,
to be solved for the unknown function y(x) with the boundary conditions
 y(0)=0, \ y(\pi/2)=2.
Without the boundary conditions, the general solution to this equation is
 y(x) = A \sin(x) + B \cos(x).\,
From the boundary condition y(0)=0 one obtains
 0 = A \cdot 0 + B \cdot 1
which implies that B=0. From the boundary condition y(\pi/2)=2 one finds
 2 = A \cdot 1
and so A=2. One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is
 y(x)=2\sin(x). \,
Types of boundary value problems
If the boundary gives a value to the normal derivative of the problem then it is a Neumann boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.
If the boundary gives a value to the problem then it is a Dirichlet boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.
If the boundary has the form of a curve or surface that gives a value to the normal derivative and the problem itself then it is a Cauchy boundary condition.
Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an elliptic operator, one discusses elliptic boundary value problems. For an hyperbolic operator, one discusses hyperbolic boundary value problems. These categories are further subdivided into linear and various nonlinear types.
Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing. Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics. The tradition of algebraic statistics In the past, statisticians have used algebra to advance research in statistics. Some algebraic statistics led to the development of new topics in algebra and combinatorics, such as association schemes. Design of experiments For example, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments. Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association schemes by R. C. Bose. Orthogonal arrays were introduced by C. R. Rao also for experimental designs. Algebraic analysis and abstract statistical inference Invariant measures on locally compact groups have long been used in statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics. Encompassing previous results on probability theory on algebraic structures, Ulf Grenander developed a theory of "abstract inference". Grenander's abstract inference and his theory of patterns are useful for spatial statistics and image analysis; these theories rely on lattice theory. Partially ordered sets and lattices Partially ordered vector spaces and vector lattices are used throughout statistical theory. Garrett Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem. Birkhoff's results have been used for maximum entropy estimation (which can be viewed as linear programming in infinite dimensions) by Jonathan Borwein and colleagues. Vector lattices and conical measures were introduced into statistical decision theory by Lucien Le Cam. Recent work using commutative algebra and algebraic geometry In recent years, the term "algebraic statistics" has been used more restrictively, to label the use of algebraic geometry and commutative algebra to study problems related to discrete random variables with finite state spaces. Commutative algebra and algebraic geometry have applications in statistics because many commonly used classes of discrete random variables can be viewed as algebraic varieties. Introductory example Consider a random variable X which can take on the values 0, 1, 2. Such a variable is completely characterized by the three probabilities p_i=\mathrm{Pr}(X=i),\quad i=0,1,2 and these numbers clearly satisfy \sum_{i=0}^2 p_i = 1 \quad \mbox{and}\quad 0\leq p_i \leq 1. Conversely, any three such numbers unambiguously specify a random variable, so we can identify the random variable X with the tuple (p0,p1,p2)âˆˆR3. Now suppose X is a Binomial random variable with parameter p = 1 âˆ’ q and n = 2, i.e. X represents the number of successes when repeating a certain experiment two times, where each experiment has an individual success probability of q. Then p_i=\mathrm{Pr}(X=i)={2 \choose i}q^i (1q)^{2i} and it is not hard to show that the tuples (p0,p1,p2) which arise in this way are precisely the ones satisfying 4 p_0 p_2p_1^2=0.\ The latter is a polynomial equation defining an algebraic variety (or surface) in R3, and this variety, when intersected with the simplex given by \sum_{i=0}^2 p_i = 1 \quad \mbox{and}\quad 0\leq p_i \leq 1, yields a piece of an algebraic curve which may be identified with the set of all 3state Bernoulli variables. Determining the parameter q amounts to locating one point on this curve; testing the hypothesis that a given variable X is Bernoulli amounts to testing whether a certain point lies on that curve or not.
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Answers:From a previous answer you have the answer to part i) of your question, namely the frequency distribution table: 3135 3 3640 9 4145 15 4650 17 5055 6 5560 0 The data was grouped into six bins (classes). The first class is from 31 to 35 minutes and there are 3 students in that bin. Therefore the frequency is 3. It appears that the data were rounded to the nearest minute and so a time 30.5 minutes would be rounded to 31 minutes. Similarly, any time less than 35.5 minutes but equal to or greater than 34.5 minutes would be rounded to 35 minutes. Therefore that first class really includes values from 30.5 to 35.5 (called the class interval). The next class is really from 35.5 to 40.5, and so on. In part (ii) you draw a histogram with class boundaries 30.5, 35.5, 40.5, 45.5 and so on. See the source below for the explanation. In part (iii) you plot a graph (a frequency polygon) by plotting the frequencies at the midpoint of each class. For example, the first point is plotted at frequency = 3 on the yaxis and time = 33 minutes on the xaxis.
Answers:When you have your class sets ... your highest and lowest classes have your highest and lowest numbers as your LCL.... (i.e. class one LCL = 11.86, and your class seven LCL = 12.1) try different ways of setting up the class boundaries so that each item gets spaced in..... EX. (your classes are set up like this) 11.86  11.89 11.90  11.93 11.94  11.97 ... and so on.... TRY messing with your class sets, since your class seven has 3 extra numbers that aren't even being used... 11.85  11.88 11.89  11.92 11.93  11.96 so on and so forth until you see that your frequencies match up with the book's ______________________________________________ Edit: I just spent time to do this for setting up class lists with the numbers you listed. If you lower each LCL and UCL by .01, then your answers will match up with the book. 11.85  11.88 11.89  11.92 11.93  11.96 11.97  12.00 12.01  12.04 12.05  12.08 12.08  12.11 If you set up those as your class lists, your frequencies will match up with the book.
Answers:b. Gender Nominal, discrete c. Sales volume of MP3 players discrete, ratio d.Soft drink preference ordinal, discrete e.Temperature continuous , interval f. SAT scores ratio , discrete (I am assuming the scores are integers (rounded if necessary) g.Student rank in class ordinal, discrete h. ratio [ ratings are scaled from 1 to 10 or some other number?] ordinal [ ratings are poor, moderate, good, excellent etc] i.Number of home computers ratio, discrete 2) a. Student IQ ratings ratio. b. Distance students travel to class.continuous c. Student scores on the first statistics test.discrete d. A classification of students by state of birth. nominal e. A ranking of students as freshman, sophomore, junior, and senior.  ordinal f. Number of hours students study per week.  ratio 3. What is the level of measurement for these items related to the newspaper business? a. The number of papers sold each Sunday during 2006.  ratio b. The departments, such as editorial, advertising, sports, etc.  nominal c. A summary of the number of papers sold by county.  ratio d. The number of years with the paper for each employee.  ratio 4. For each of the following, determine whether the group is a sample or a population. a. The participants in a study of a new cholesterol drug.  sample b. The drivers who received a speeding ticket in Kansas City last month.  population c. Those on welfare in Cook County (Chicago), Illinois.  population d. The 30 stocks reported as a part of the Dow Jones Industrial Average.  sample
Answers:It is difficult to know what "an original example" means. They will all have occurred, or been used before. However, the one which always springs to mind as an example of type 2 error is someone being given a blood test for a particular disease, say, and being told that the result is negative when they do, in fact, actually have the disease. Not very original, but I don't think there are any original ones to be had !