统计学(statistics).ppt

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1、Chapter 1Introduction統計學統計學(statistics)統計學是指搜集、整理、表現、分析解釋資料，並藉科學的統計推論方法，在不確定的情況下，由樣本資料所獲得的結果，來推論母體的性質與事實，從而做出適切決策的一門學科。1-1林惠玲林惠玲 陳正倉著陳正倉著 雙葉書廊發行雙葉書廊發行 2000母體與樣本母體與樣本第1章 緒論 3統統計計學圖圖1.1 母體與樣本間的關係母體與樣本間的關係林惠玲林惠玲 陳正倉著陳正倉著 雙葉書廊發行雙葉書廊發行 20001-1林惠玲林惠玲 陳正倉著陳正倉著 雙葉書廊發行雙葉書廊發行 2000母體參數與樣本統計量母體參數與樣本統計量第1章 緒論 5統

2、統計計學1-1第1章 緒論 6統統計計學林惠玲林惠玲 陳正倉著陳正倉著 雙葉書廊發行雙葉書廊發行 1999隨機實驗隨機實驗(random experiment)(random experiment)1-1林惠玲林惠玲 陳正倉著陳正倉著 雙葉書廊發行雙葉書廊發行 1999隨機實驗的基本觀念隨機實驗的基本觀念表表5.1 隨機實驗、出象與樣本空間隨機實驗、出象與樣本空間林惠玲林惠玲 陳正倉著陳正倉著 雙葉書廊發行雙葉書廊發行 1999examplesEx1.1-1:Two dice are cast and the total number of spots on the sides that ar

3、e“up”are counted.The outcome spaces is S=2,3,4,5,6,7,8,9,10,11,12.Ex1.1-2:Each of the six students selects an integer at random from the first 52 positive integers.We are interested in whether at least two of these six integers match(M)or whether all different(D).Thus S=M,D.Ex1.1-3:A fair coin is fl

4、ipped successively at random until the first head is observed.If we let x denote the number of flips of the coin that are required,then S=x :x=1,2,3,4,which consists of an infinite,but countable,number of outcomes.Ex1.1-4:A box of breakfast cereal contains one of four different prizes.The purchase o

5、f one box of cereal yields one of the prizes as the outcome,and the sample spaces is the set of four different prizes.Ex1.1-5:In Example 1.1-4,assume that the prizes are put into the boxes randomly.A family continues to buy this cereal until they obtain a complete set of the four different prizes.Th

6、e number of boxes cereal that must be purchased is one of the outcomes in S=b :b =4,5,6,.Ex1.1-6:A fair coin is flipped successively at random until heads is observed on two successively flips.If we let y denote the number of flips of the coin that are required,then S=y:y=2,3,4,.Ex1.1-7:To determine

7、 the percentages of body fat for a person,one measurement that is made is a persons weight under water.If denotes this weight in kilograms,then the sample spaces could be S=:,as we know from past experience that this weight does not exceed 7 kilograms.Ex1.1-8:An ornithologist is interested in the cl

8、utch size(number of eggs in a nest)for gallinules,a bird that lives in a marsh.Then a possible sample spaces would be S=0,1,15,as 15 is the largest known clutch size.隨機試驗的結果,可能是數字,也可能是文字或符號.隨機試驗的結果,可能是有限個,例如1.1-1,1.1-4,1.1-8;可能是無限個,但可數,例如1.1-3,1.1-6;也可能是不可數,例如1.1-7.Remark:隨機實驗的基本觀念隨機實驗的基本觀念圖圖5.2 簡單事

9、件與複合事件簡單事件與複合事件次數分配表次數分配表(frequency table)&(frequency table)&相對次數直方圖相對次數直方圖(relative frequency histogram)機率質量函數機率質量函數(probability mass function)當隨機試驗的次數n很小,相對次數f/n非常不穩定,但是當n增加時,它趨近於某個數p(稱為機率值).若以h(x)表示各種不同x值的相對次數,則我們稱這個函數f(x)為機率質量函數(probability mass function)簡稱為p.m.f.機率質量函數機率質量函數(probability mass fu

10、nction)假設我們有某一函數f(x),為一隨機試驗結果的機率模型.如果我們重複試驗很多次,我們預期所得到之相對次數直方圖h(x)將趨近於f(x).實際上,沒有一個模型是全然正確的;但很接近正確的模型,在實務是非常有用的.這學期,我們將學習機率觀念,以便將來為各個案例選擇某個合理的機率模型;下學期,我們將學習模型檢測及模型參數估計.Ex1.1-10 Let us place six chips of the same size in a bowl.The number one is placed on each of three of the chips,the number two on

11、each of two of the other chips,and finally the number three on the last chip.The experiment is to select one chip at random and read the number x on the chip.Here S=1,2,3.The model is given by the p.m.f.We simulated this experiment using a computer n=600 times obtaining the respective frequencies 28

12、4,208,108.隨機變數隨機變數(random variable)(random variable)之定義之定義Definition 3.1-1:Given a random experiment with an outcome space S,a function X that assigns to each element s in S one and only one real number X(s)=x is called a random variable.The space of X is the set of real number Ex3.1.1:A rat is sele

13、cted at random from a cage and its sex is determined.The set of possible outcomes is female and male.Thus the outcome space is S=female,male =F,M.Let X be a function defined on S such that X(F)=0 and X(M)=1.Random variable:1.disrcrete type 2.continuous type母體平均數母體平均數(mean),(mean),變異數變異數(variance)&(v

14、ariance)&標準差標準差(standard deviation)(standard deviation)Suppose the random variable X has the space And these points have respective probabilities Where f(x)is the p.m.f.Of course,Define the mean of the random variable X by And the second moment about the mean is defined by is called the variance of

15、the random variable X.The positive square root of the variance is called the standard deviation of X and is denote by the sigma having symbol .the variance is .Note that the variance can be computed in another way,becauseEx 1.2-1 Let X equal the number of spots on the side that is up after a six-sid

16、ed(one of a pair of dice)is cast at random.If everything is fair about this experiment,a reasonable probability model is given by the p.m.f.The mean of X is The second moment about the origin isThus the variance equalsThe standard deviation is 樣本平均數樣本平均數(sample mean),樣本變異數樣本變異數(sample variance)令令 是樣

17、本觀測值是樣本觀測值定義樣本平均數定義樣本平均數 樣本變異數樣本變異數Ex 1.2-2 Rolling a fair six-sided die five times could result in the following sample of n=5 observations.In this case andIt follows that Continuous-type dataThe observation values come from an interval of possible outcomes.Ex:(1)the length of time of waiting in li

18、ne to buy tickets.(2)the weight of a“one pound“package of hot dogs could be any number between 0.94 and 1.25 pounds.Ex13-1:The weights in grams of 40 miniature Baby Ruth candy bars,with the weights ordered,are given in Table 1.3-1.20.5 20.7 20.8 22.522.6 22.6 22.7.23.523.6 23.6 23.6.24.824.9 24.9 25

19、.1.26.7全距26.7-20.5=6.3 Divide the interval(20.45,26.75)into k=7 classes of width 0.9,then construct the frequency table and relative frequency histogram.Here,the relative frequency histogram is constructed by drawing a rectangle for each class with base length=height=as n andas the lengths of the cl

20、ass intervals then h(x)some function,say f(x)X:random variable of conti-type the relative frequency of the interval(a,b)probability of the event A=x:axb P(A)P(aX0 for all(b)(c)the probability of the event A=x:axb is Ex1.7-1 Suppose a balanced spinner is constructed so that the result of a spin is a

21、random variable X whose space is S=x:0 x1 X has the p.d.f f(x)=1,0 x1 P(0.25X0.75)=P(0.5X0.6)=P(0.1X0.4)=The comparison of the probability distribution for discrete&continuous case discrete contiSpace S=an interval (or union of intervals)p.d.fMeanVarianceEx.1.7-2:If Y is the largest of two spins of

22、the balanced spinner of Ex.17.1,Y has the p.d.f f(y)=2y,0 y 1 where the space of Y is S=y:0y1 the mean of Y is連續型分配的百分位數連續型分配的百分位數(a percentile of a distribution of continuous type)Def:Let X be a continuous type random variable with p.d.f f(x).If 0p1,the(100p)th percentile is a number p such that th

23、e area under f(x)to the left of pis p.That isThe 50th percentile 0.5 is called the median(中位數)The 25th percentile 0.25 is called the first quartile(第一四分位數)The 75th percentile 0.75 is called the third quartile(第三四分位數)連續型分配的百分位連續型分配的百分位(a percentile of a distribution of continuous type)Ex.1.7-6Let us

24、consider the distribution of the largest value,Y,of two spins of the balanced spinner of Ex.1.7-2 with p.d.f f(y)=2y,0 y1.樣本百分位樣本百分位(sample percentile)For a sample of n observations,Ordered the observations from small to large,say ,the resulting ordered data are called order statistics of the sample

25、.If 0p1,the(100p)th sample percentile,denoted byhas approximately np sample observations less than it and also n(1-p)sample observations greater than it.樣本百分位樣本百分位(sample percentile)One way to get the sample percentile is as following:(1)若(n+1)p是整數,則將第(n+1)p個順序統計量視為第100p個樣本百分位數.(2)若(n+1)p不是整數,say ,w

26、here 0a/b1,.我們定義第100p個樣本百分位數為(3)pn/n+1,則樣本百分位數未被定義.Ex:考慮50個測驗排序分數,Median=第50個百分位數First quartile=第25個百分位數Third quartile=第75個百分位數箱線圖(box-and-whisker diagram)顯示minimum,first quartile,median,third quartile,and maximum的圖形.q-q plotLet 是隨機樣本 的順序統計量,如果理論分配是觀測值的一個好模型,則這些點 將接近一條通過原點,且斜率為的直線,因為 .對各個不同的r所形成的 圖被稱為 q-q plot(quartile-quartile plot).