概率論與數(shù)理統(tǒng)計(jì)英文版總結(jié)

1、samplespace樣本空間the set of all possible outcomes of a statistical experiment is called the sample space.event 事件an event is a subset of a sample space. certain event（必然事件）：the sample space itself, is certainly an event, which is called a certain event, means that it always occurs in the experiment. i

2、mpossible event（不可能事件）：the empty set, denoted by, is also an event, called an impossible event, means that it never occurs in the experiment. probability of events （概率）if the number of successes in trails is denoted by , and if the sequence of relative frequencies obtained for larger and larger valu

3、e of approaches a limit, then this limit is defined as the probability of success in a single trial.“equally likely to occur”-probability（古典概率） if a sample space consists of sample points, each is equally likely to occur. assume that the event consists of sample points, then the probability that a o

4、ccurs is mutually exclusive(互斥事件)definition 2.4.1 events are called mutually exclusive, if .theorem 2.4.1 if and are mutually exclusive, then (2.4.1) mutually independent 事件的獨(dú)立性 two events and are said to be independent if or two events and are independent if and only if .conditional probability 條件概

5、率the probability of an event is frequently influenced by other events. definition the conditional probability of , given , denoted by , is defined by if . (2.5.1)the multiplication theorem乘法定理 if are events, then if the events are independent, then for any subset , (全概率公式 total probability)theorem 2

6、.6.1. if the events constitute a partition of the sample space s such that for than for any event of , (2.6.2)（貝葉斯公式bayes formula.）theorem 2.6.2 if the events constitute a partition of the sample space s such that for than for any event a of s, , . for (2.6.2)proof by the definition of conditional p

7、robability, using the theorem of total probability, we have 1. random variable definitiondefinition 3.1.1 a random variable is a real valued function defined on a sample space; i.e. it assigns a real number to each sample point in the sample space.2. distribution functiondefinition 3.1.2 let be a ra

8、ndom variable on the sample space . then the function . is called the distribution function of note the distribution function is defined on real numbers, not on sample space.3. propertiesthe distribution function of a random variable has the following properties:(1) is non-decreasing.in fact, if , t

9、hen the event is a subset of the event ,thus (2), .(3)for any , .this is to say, the distribution function of a random variable is right continuous.3.2 discrete random variables 離散型隨機(jī)變量definition 3.2.1 a random variable is called a discrete random variable, if it takes values from a finite set or, a

10、 set whose elements can be written as a sequence geometric distribution （幾何分布） x 1234kppq1pq2pq3pqk1pbinomial distribution（二項(xiàng)分布）definition 3.4.1 the number of successes in bernoulli trials is called a binomial random variable. the probability distribution of this discrete random variable is called t

11、he binomial distribution with parameters and , denoted by .poisson distribution（泊松分布）definition 3.5.1 a discrete random variable is called a poisson random variable, if it takes values from the set , and if , (3.5.1)distribution (3.5.1) is called the poisson distribution with parameter, denoted by .

12、expectation (mean) 數(shù)學(xué)期望definition 3.3.1 let be a discrete random variable. the expectation or mean of is defined as (3.3.1)2variance 方差 standard deviation （標(biāo)準(zhǔn)差）definition 3.3.2 let be a discrete random variable, having expectation . then the variance of , denote by is defined as the expectation of t

13、he random variable (3.3.6)the square root of the variance , denote by , is called the standard deviation of : (3.3.7) probability density function 概率密度函數(shù)definition 4.1.1 a function f(x) defined on is called a probability density function （概率密度函數(shù)）if:(i) ;(ii) f(x) is intergrable (可積的) on and .definit

14、ion 4.1.2 let f(x) be a probability density function. if x is a random variable having distribution function , (4.1.1)then x is called a continuous random variable having density function f(x). in this case,. (4.1.2) 5. mean（均值）definition 4.1.2 let x be a continuous random variable having probabilit

15、y density function f(x). then the mean (or expectation) of x is defined by, (4.1.3)provided the integral converges absolutely. 6. variance方差similarly, the variance and standard deviation of a continuous random variable x is defined by, (4.1.4)where is the mean of x, is referred to as the standard de

16、viation.we easily get. (4.1.5).4.2 uniform distribution 均勻分布the uniform distribution, with the parameters a and b, has probability density function4.5 exponential distribution 指數(shù)分布definition 4.5.1 a continuous variable x has an exponential distribution with parameter , if its density function is giv

17、en by (4.5.1)theorem 4.5.1 the mean and variance of a continuous random variable x having exponential distribution with parameter is given by.4.3 normal distribution 正態(tài)分布1. definitionthe equation of the normal probability density, whose graph is shown in figure 4.3.1, is4.4 normal approximation to t

18、he binomial distribution（二項(xiàng)分布）, n is large (n30), p is close to 0.50,4.7 chebyshevs theorem（切比雪夫定理）theorem 4.7.1 if a probability distribution has mean and standard deviation , the probability of getting a value which deviates from by at least k is at most . symbolically , .joint probability distrib

19、ution（聯(lián)合分布）in the study of probability, given at least two random variables x, y, ., that are defined on a probability space, the joint probability distribution for x, y, . is a probability distribution that gives the probability that each of x, y, . falls in any particular range or discrete set of

20、values specified for that variable.5.2 conditional distribution 條件分布 consistent with the definition of conditional probability of events when a is the event x=x and b is the event y=y, the conditional probability distribution of x given y=y is defined as for all x provided .5.3 statistical independe

21、nt 隨機(jī)變量的獨(dú)立性definition 5.3.1 suppose the pair x, y of real random variables has joint distribution function f(x,y). if the f(x,y) obey the product rule for all x,y.the two random variables x and y are independent, or the pair x, y is independent.5.4 covariance and correlation 協(xié)方差和相關(guān)系數(shù)we now define tw

22、o related quantities whose role in characterizing the interdependence of x and y we want to examine.definition 5.4.1 suppose x and y are random variables. the covariance of the pair x,y is .the correlation coefficient of the pair x, y is.where definition 5.4.2 the random variables x and y are said t

23、o be uncorrelated iff . 5.5 law of large numbers and central limit theorem 中心極限定理we can find the steadily of the frequency of the events in large number of random phenomenon. and the average of large number of random variables are also steadiness. these results are the law of large numbers.theorem 5

24、.5.1 if a sequence of random variables is independent, with then. (5.5.1)theorem 5.5.2 let equals the number of the event a in n bernoulli trials, and p is the probability of the event a on any one bernoulli trial, then. (5.5.2)(頻率具有穩(wěn)定性)theorem 5.5.3 if is independent, withthen . population (總體)defi

25、nition 6.2.1 a population is the set of data or measurements consists of all conceivably possible observations from all objects in a given phenomenon. .a population may consist of finitely or infinitely many varieties. sample （樣本、子樣）definition 6.2.2 a sample is a subset of the population from which

26、people can draw conclusions about the whole.sampling(抽樣)taking a sample: the process of performing an experiment to obtain a sample from the population is called sampling. 中位數(shù)definition 6.2.4 if a random sample has the order statistics , then(i) the sample median is (ii) the sample range is .sample

27、distributions 抽樣分布1sampling distribution of the mean 均值的抽樣分布theorem 6.3.1 if is mean of the random sample of size from a random variable which has mean and the variance , then and .it is customary to write as and as . here, is called the expectation of the mean.均值的期望 is called the standard error of

28、the mean. 均值的標(biāo)準(zhǔn)差7.1 point estimate 點(diǎn)估計(jì)definition 7.1.1 suppose is a parameter of a population, is a random sample from this population, and is a statistic that is a function of . now, to the observed value , if we use as an estimated value of , then is called a point estimator of and is referred as

29、a point estimate of . the point estimator is also often written as .unbiased estimator(無(wú)偏估計(jì)量)definition 7.1.2. suppose is an estimator of a parameter . then is unbiased if and only if minimum variance unbiased estimator（最小方差無(wú)偏估計(jì)量）definition 7.1.3 let be an unbiased estimator of . if for any which is

30、 also an unbiased estimator of , we have,then is called the minimum variance unbiased estimator of . sometimes it is also called best unbiased estimator.3. method of moments 矩估計(jì)的方法definition 7.1.4 suppose constitute a random sample from the population x that has k unknown parameters . also, the popu

31、lation has firs k finite moments that depends on the unknown parameters. solve the system of equations, (7.1.4)to get unknown parameters expressed by the observations values, i.e. for . then is an estimator of by method of moments. definition7.2.1 suppose that is a parameter of a population, is a ra

32、ndom sample of from this population, and and are two statistics such that . if for a given with , we have.then we refer to as a confidence interval for . moreover, is called the degree of confidence. and are called lower and upper confidence limits. the estimation using confidence interval is called interval estimation. confidence interval- 置信區(qū)間 lower confidence limits- 置信下限 upper confidence limits- 置信上限degree of confidence-置信度2極大似然函數(shù)likelihood function definition 7.5.1 a random sample has the observed