(完整版)概率论与数理统计英文版第三章

1、Chapter 3.Ran dom Variables and Probability Distributi on1. Concept of a Random VariableExample : three electronic components are testedsample space (N: non defective, D: defective)S =NNN, NND, NDN, DNN, NDD, DND, DDN, DDDallocate a nu merical descripti on of each outcomeconcerned with the nu mber o

2、f defectiveseach point in the sample space will be assig ned a nu merical value of 0, 1,2, or 3.ran dom variable X: the nu mber of defective items, a ran dom qua ntityran dom variableDefin iti on 3.1A random variable is a function that associates a real number with each element in the sample space.X

3、: a ran dom variablex : one of its valuesEach possible value of Xreprese nts an event that is a subset of the sample spaceelectr onic comp onent test:E =DDN, DND, NDD=X = 2.Example 3.1 Two balls are draw n in successi on without replaceme nt from an urn containing 4 red balls and 3 black balls. Y is

4、 thenumber of red balls. The possible outcomes and the values y of the random variable Y ?Example 3.2 A stockroom clerk returns three safety helmets at random to three steel mill employees who had previously checkedthem. If Smith, Jones, and Brown, in that order, receive one of the three hats, list

5、the sample points for the possible orders of returning the helmets,a nd find the value m of the ran dom variable M that represe nts the nu mber of correct matches.The sample space contains a fin ite nu mber of eleme nts in Example 3.1 and 3.2.ano ther example: a die is throw n un til a 5 occurs,F: t

6、he occurre nee of a 5N: the nono ccurre nee of a 5obta in a sample space with an unending seque nee of eleme ntsS =F, NF, NNF, NNNF, . . .the nu mber of eleme nts can be equated to the nu mber of whole nu mbers; can be coun tedDefinition 3.2If a sample space contains a finite number of possibilities

7、 or an unending sequence with as many elements asthere are whole numbers, it is called a discrete sample space .The outcomes of some statistical experime nts may be n either fin ite nor coun table.example: measure the distances that a certain make of automobile will travel over a prescribed test cou

8、rse on 5 liters of gasoli nedista nce: a variable measured to any degree of accuracy we have infinite number of possible distances in the sample space, cannotbe equated to the number of whole numbers .Defin iti on 3.3If a sample space contains an infinite number of possibilities equal to the number

9、of points on a line segment, it is calledacontinu ous sample spaceA random variable is called a discrete random variableif its set of possible outcomes is countable.Y in Example 3.1 and M in Example 3.2 are discrete ran dom variables.When a random variable can take on values on a continuous scale, i

10、t is called a continuous random variable .The measured dista nee that a certa in make of automobile will travel over a test course on 5 liters of gasoli ne is a continu ous random variable.continuous random variables represent measured data :all possible heights, weights, temperatures, dista nee, or

11、 life periods.discrete random variables represent count data: the number of defectives in a sample of k items, or the number of highway fatalitiesper year in a give n state.2.Discrete Probability Distributio nA discrete random variable assumes each of its values with a certain probabilityassume equa

12、l weights for the elements in Example 3.2, whats the probability that no employee gets back his right helmet. Theprobability that M assumed the value zero is 1/3.The possible values m of M and their probabilities are0131/31/21/6Probability Mass FunctionIt is convenient to represent all the probabili

13、ties of a random variable X by a formula.write p(x) = P (X = x)The set of ordered pairs (x, p(x) is called the probability function or probability distributionof the discrete random variableX.Defin iti on 3.4The set of ordered pairs (x, p(x) is a probability function, probability mass function, or p

14、robability distributi on of the discrete ran domvariable X if, for each possible outcome x心)X。 Ez ”3)= i3P(X= x) = p(a?)Example 3.3 A shipme nt of 8 similar microcomputers to a retail outlet contains 3 that are defective. random purchase of 2 of thesecomputers, find the probability distribution for

15、the number of defectives. Soluti onX: the possible nu mbers of defective computersx can be any of the nu mbers 0, 1, and 2.I L)MO)弓FfX= 0)=十产=p(l) = p(,V = l) p(2) = r( (.V = 2)Cumulative FunctionThere are many problem where we may wish to compute the probability that the observed value of a random

16、variable X will be lesstha n or equal to some real nu mber x.Writ ing F (x) = P (Xx) for every real nu mber x.If a school makes aDefin iti on 3.5The cumulative distributi onF (x) of a discrete ran dom variable X with probability distributio n p(x) is= P(.V i) =for x z For the ran dom variable M, the

17、 nu mber of correct matches in Example 3.2, we haveF(2)=尸刖 2j =再叫二 I-i =-.J 3&The cumulative distributi on of M is冷for in 01/3for 0 w 1F(m) f5/6for 1 3.Remark. the cumulative distributi on is defi ned not only for the values assumed by give nran dom variable but for all real nu mbers.Example 3.5

18、 The probability distributi on of X is01234_L13丄丄16484Find the cumulative distribution of the random variable X.Certain probability distribution are applicable to more than one physical situation.The probability distribution of Example 3.5 can apply to different experimental situations.Example 1:the

19、 distribution of Y , the number of heads when a coin is tossed 4 timesExample 2:the distribution of W , the number of read cards that occur when 4 cards are drawn at random from a deck insuccessi on with each card replaced and the deck shuffled before the next draw ing. graphsIt is helpful to look a

20、t a probability distribution in graphic form. bar chart;histogram; cumulative distributi on.3. Continuous Probability DistributionContinuous Probability distributionA continuous random variable has a probability of zero of assuming exactly any of its values. Consequently, its probability distributio

21、nI)for 2- (11/ICfor 0 j 15/1Ufor 1 a- 2ll/lfifor 2 r 315/(6for 3 ；r 4.cannot be given in tabular form.Example : the heights of all people over 21 years of age (random variable)Between 163.5 and 164.5 centimeters, or even 163.99 and 164.01 centimeters, there are an infinite number of heights, one of

22、whichis 164 cen timeters.The probability of selecti ng a pers on at ran dom who is exactly 164 cen timeters tall and not one of the infin itely large set of heightsso close to 164 cen timeters is remote.We assign a probability of zero to a point , but this is not the case for an interval . We will d

23、eal with an interval rather than a point value,such as P (a X c).P (aXwb) = P (a Xb) = P (aX b) = P (a X OPfor all r R.2.= 13.Pa X b) = j：血.Example 3.6 Suppose that the error in the react ion temperature, in oC, for a con trolled laboratory experime nt is a con ti nu ous random variable X hav ing th

24、e probability den sity function(-1 r 2= tI 0.elsewhere.(a) Verify condition 2 of Definition 3.6.(b) Find P (0 Xw1).Solution :.P (0 Xw1) = 1/9.Defin iti on 3.7The cumulative distributi onF (x) of a con ti nu ous ran dom variable X with den sity function f(x) isF(i) = P(X x) = ffor oc JLimmediate con

25、seque nee:X b) = F(b) - F(a)andf(x) =Example 3.7 For the den sity function of Example 3.6 find F (x), and use it toevaluate P (0 x1).= -0.-1 x 2elsewhere.j -1-1 x 2 z 0. For allJ?.i) ).2爲心小=I-3.A = Y=曲=p(jr. .For region A in the rtf plane zl lit就嘗Example 3.8Two refills for a ballpo int pen are selec

26、ted at ran dom from a box that contains 3 blue refills,2 red refills, and 3 gree n refills. If X is thenumber of blue refills and Y is the number of red refills selected, find(a) the joint probability fun cti on p(x, y)(b) P (X, Y )A where A is the region(x, y)|x + y 0 for allCT.y)F2* J二丄二g 心血咖=tM P

27、(X. Y)e A =U)dxdyfor any regron A in the xy plane.Whe n X and Y are continu ous ran dom variables, the joint den sity function f(x, y) is a surface lying above the xy pla ne.P (X, Y )A, where A is any region in the xy plane, is equal to the volume of the right cylinder bounded by the base A and thes

28、urface.Example 3.9Suppose that the joint den sity fun cti on ismarginal distributionp (x, y): the joint probability distribution of the discrete random variables X and Ythe probability distribution p X(x) of X alone is obtained by summing p(x, y) over the values of Y .Similarly, the probability dist

29、ribution p Y (y) of Y alone is obtained by summing p(x, y) over the values of X. pX (x) and p Y (y):marginal distributi onsof X and YWhe n X and Yare con ti nu ous ran dom variables, summatio ns are replaced by in tegrals.Defin iti on 3.10 The margi nal distributi on of X alone and of Y alone arePE

30、= Vand 阿仙卜= Example 3.10Show that the column and row totals of Table3.1 give the marginal distribution of X alone and of Y alone.prese nt the results in Table 3.1Idswhere.(a)Verifyconditioni 2 of Definition 39(b) Find P (Xifor the discrete case, and byfor the coritiTiuoLJ5iExample 3.11 Fi nd margina

31、l probability den sity functions fx(x) and fy(y)for the joint density function of Example 3.9./ 2：2訟+3必0 lt0 / 1)=5elsewhere.The marginal distributi on px(x) or fx(x) and px(y) or fy(y) are in deed the probability distributi on of the in dividual variable X andY , respectively.How to verify?The cond

32、itions of Definition 3.4 or Definition 3.6 are satisfied.Conditional distribution recall the defi niti on of con diti onal probability:X and Y are discrete ran dom variables, we haveThe value x of the random variable represent an event that is a subset of the sample space.Defin iti on 3.11Let X and

33、Y be two discrete ran dom variables.The con diti onal probability mass fun ctio n of the ran dom variable Y , give nthat X = x, isSimilarly, the conditional probability mass function of the random variable X, given that Y = y, isfx=Defin iti on 3.11Let X and Y be two con ti nu ous ran dom variables. The con diti onal probability den sity fun cti on give nthat X = x, isSimilarly, the conditional probability density function of the random variable X, given that YRemark:The function f(x, y)/ fX(x) is strictly a function of