数学专业英语12.docx

1、数学专业英语12Mathematical EnglishDr. Xiaomin ZhangEmail: zhangxiaomin2.12 Probability Theory and Mathematical StatisticsTEXT A Special terminology peculiar to probability theoryIn discussions involving probability, one often sees phrases from everyday language such as “two events are equally likely,” “an

2、 event is impossible,” or “an event is certain to occur.” Expressions of this sort have intuitive appeal and it is both pleasant and helpful to be able to employ such colorful language in mathematical discussions. Before we can do so, however, it is necessary to explain the meaning of this language

3、in terms of the fundamental concepts of our theory.Because of the way probability is used in practice, it is convenient to imagine that each probability space (S, B, P) is associated with a real or conceptual experiment. The universal set S can then be thought of as the collection of all conceivable

4、 outcomes of the experiment, as in the example of coin tossing discussed in the foregoing section. Each element of S is called an outcome or a sample and the subsets of S that occur in the Boolean algebra B are called events. The reasons for this terminology will become more apparent when we treat s

5、ome examples.Assume we have a probability space (S, B, P) associated with an experiment. Let A be an event, and suppose the experiment is performed and that its outcome is x. (In other words, let x be a point of S.) This outcome x may or may not belong to the set A. If it does, we say that the event

6、 A has occurred. Otherwise, we say that the event A has not occurred, in which case xA, so the complementary event A has occurred. An event A is called impossible if A=, because in this case no outcome of the experiment can be an element of A. The event A is said to be certain if A=S, because then e

7、very outcome is automatically an element of A.Each event A has a probability P(A) assigned to it by the probability function P. (The actual value of P(A) or the manner in which P(A) is assigned is not concern us at present.) The number P(A) is also called the probability that an outcome of the exper

8、iment is one of the elements of A. We also say that P(A) is the probability that the event A occurs when the experiment is performed.The impossible event must be assigned probability zero because P is finitely additive measure. However, there may be events with probability zero that are not impossib

9、le. In other words, some of the nonempty subsets of S may be assigned probability zero. The certain event S must be assigned probability 1 by the very definition of probability, but there may be other subsets as well that are assigned probability 1. In example 1 of Section 6.8 there are nonempty sub

10、sets with probability zero and proper subsets of S that have probability 1.Two events A and B are said to be equally likely if P(A)=P(B). The event A is called more likely than B if P(A)P(B), and at least as likely as B if P(A)P(B). Table 2-12-1 provides a glossary or further everyday language that

11、is often used in probability discussions. The letters A and B represent events, and x represents an outcome of an experiment associated with the sample space S. Each entry in the left-hand column is a statement about the events A and B, and the corresponding entry in the right-hand column defines th

12、e statement in terms of set theory.Notationsprobability function here the value of probability function P at point A is the probability that the event A occurs. Generally, The probability function P(x) (also called the probability density function or density function) of a continuous distribution is

13、 defined as the derivative of the (cumulative) distribution function D(x), so A probability function satisfies and is constrained by the normalization condition, Special cases are To find the probability function in a set of transformed variables, find the Jacobian. For example, If u=u(x), then so S

14、imilarly, if u=u(x, y) and v=v(x, y), then Given n probability functions P1(x), P2(x), ., Pn(x), the sum distribution X+Y+Z has probability function where (x) is a delta function. Similarly, the probability function for the distribution of XYZ is given by The difference distribution X-Y has probabil

15、ity function and the ratio distribution X/Y has probability function TEXT B two basic statistics concepts population and sampleIn the preceding sections, we cited a few examples of situations where evaluation of factual information is essential for acquiring new knowledge. Although these examples ar

16、e drawn from widely differing fields and only sketchy descriptions of the scope and objectives of the studies are provided, a few common characteristics are readily discernible.First, in order to acquire new knowledge, relevant date must be collected. Second, some amount of variability in the data i

17、s unavoidable even though observations are made under the same or closely similar conditions. For instance, the treatment for an allergy may provide long-lasting relief for some individuals whereas it may bring only transient relief or even none at all to others. Likewise, it is unrealistic to expec

18、t that college freshmen whose high school records were alike would perform equally well in college. Nature does not follow such a rigid law. A third notable feature is that access to a complete set of data is either physically impossible or from a practical standpoint not feasible. When data are obt

19、ained from laboratory experiments or field trials, no matter how much experimentation has been performed, more can always be done. In public opinion or consumer expenditure studies, a complete body of information would emerge only if data were gathered from every individual in the nation undoubtedly

20、 a monumental if not impossible task. To collect an exhaustive set of data related to the damage sustained by all cars of a particular model under collision at a specified speed, every car of that model coming off the production lines would have to be subjected to a collision! Thus, the limitations

21、of time, resources, and facilities, and sometimes the destructive nature of the testing, mean that we must work with incomplete information the data that are actually collected in the course of an experimental study.The preceding discussions highlight a distinction between the data set that is actua

22、lly acquired through the process of observation and the vast collection of all potential observation that can be conceived in given context. The statistical name for the former is sample; for the latter, it is population, or statistical population. To further elucidate these concepts, we observe tha

23、t each measurement in the data as originates from a distinct source which may be a patient, tree, farm, household, or some other entity depending on the object of a study. The source of each measurement is called a sampling unit, or simply, a unit. A sample or sample data set then consists of measur

24、ements recorded for those units that are actually observed. The observed units constitute a part of a far larger collection about which we wish to make inferences. The set of measurements that would result of all the units in the larger collection could be observed is defined as the population.Defin

25、ition 1 A statistical population is the set of measurements (or record of some qualitative trait) corresponding to the entire collection of units about which information is sought.The population represents the target of an investigation. We learn about the population by taking a sample from the popu

26、lation.Definition 2 A sample from a statistical population is the set of measurements that are actually collected in the course of an investigation. SUPPLEMENT A Bertrands paradoxConsider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the pr

27、obability that the chord is longer than a side of the triangle?This problem was originally posed by Joseph Bertrand in his work, Calcul des probabilits (1888). Bertrand gave three arguments, all apparently valid, yet yielding inconsistent results. where red = longer than triangle side, blue = shorte

28、r. Selection method 1 Choose a point on the circle and rotate the triangle so that the point is at one vertex. Choose another point on the circle and draw the chord joining it to the first point. For points on the arc between the endpoints of the side opposite the first point, the chord is longer th

29、an a side of the triangle. The length of the arc is one third of the circumference of the circle, therefore the probability a random chord is longer than a side of the inscribed triangle is one third.Selection method 2 Choose a radius of the circle and rotate the triangle so a side is perpendicular

30、to the radius. Choose a point on the radius and construct the chord whose midpoint is the chosen point. The chord is longer than a side of the triangle if the chosen point is nearer the center of the circle than the point where the side of the triangle intersects the radius. Since the side of the tr

31、iangle bisects the radius, it is equally probable that the chosen point is nearer or farther. Therefore the probability is one half.Selection method 3 Choose a point anywhere within the circle and construct a chord with the chosen point as its midpoint. The chord is longer than a side of the inscrib

32、ed triangle if the chosen point falls within a concentric circle of radius 1/2. The area of the smaller circle is one fourth the area of the larger circle, therefore the probability is one fourth.Bertrand intended to show that the classical definition of probability is not applicable to a problem with an infinity of possible outcomes. According to the classical definition, the probability of a compound event is the ratio of the number of favorable cases to the total number of cases. Such a definitio