(完整版)概率论与数理统计英文版总结,推荐文档

9. Nonparametric Statistics9.1 Sign Test 符号检验1The simplest of all nonparametric methods is the sign test, which is usually used to test the significance of the difference between two means in a paired experiment.最简单的非参数检验是符号检验检验两个总体均值差的显著程度It is particularly suitable when the various pairs are observed under different conditions, a case in which the assumption of normality may not hold. However, because of its simplicity, the sign test is often used even though the populations are normally distributed. As is implied by its name in this test only the sign of the differencebetween the paired variates is used.若两个总体的均值相等，那么符号‘＋’、‘－’的概率一样。

D = sign of (X 1－X 2 )If p denotes the probability of a difference D being positive andq the probability of its being negative, we have as hypothesis p=1/2. appropriate test statistic is X , X~B (n, p)， X --- N(‘+”)we will reject 0Hin favor of1Honly if the proportion of plussigns is sufficiently less than 1/2, that is , when the value x of our random variable is small. Hence, if the computed P -value12()P P X x when p =≤=is less than or equal to the significance level α, we reject 0Hinfavor of1H .we reject0Hin favor1Hwhen the proportion of plus signs issignificantly less than or significantly greater than 1/2. This, of course, is equivalent to x being sufficiently small or sufficiently large, respectively. Therefore, if /2x n < and the computed P-value 122()P P X x when p =≤=is less than or equal to α, or if /2x n > and the computed P-value 122()P P X x when p =≥= is less than or equal to α, we reject 0Hin favor1H .Car Radial tires Belted tires D1 4.2 4.1 ＋ 2 4.7 4.9 －3 6.6 6.2 ＋4 7.0 6.9 ＋5 6.7 6.8 －6 4.5 4.4 ＋7 5.7 5.78 6.0 5.8 ＋9 7.4 6.9 ＋10 4.9 4.911 6.1 6.0 ＋12 5.2 4.9 ＋13 5.7 5.3 ＋14 6.9 6.5 ＋15 6.8 7.1 －16 4.9 4.8 ＋符号检验的利弊n 必须比较大因为对于n =5的样本，会出现永远不拒绝“总体均值相等“的假设。

Sample Space样本空间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 S itself, is certainly an event, which is called a certain event, means that it always occurs in the experiment、impossible 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 n trails is denoted by s, and if the sequence of relative frequencies /s n obtained for larger and larger value of n 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 S consists of N sample points, each is equally likely to occur、Assumethat the event A consists of n sample points, then the probability p that A occurs is()np P AN==Mutually exclusive(互斥事件)Mutually independent 事件的独立性Two events A and B are said to be independent if()()()P A B P A P B=⋅IOr Two events A and B are independent if and only if(|)()P B A P B=、Conditional Probability 条件概率The probability of an event is frequently influenced by other events、If 12k ,,,A A A L are events, then12k 121312121()()(|)(|)(|)k k P A A A P A P A A P A A A P A A A A -=⋅⋅I I L I L I I L I If the events 12k ,,,A A A L are independent, then for any subset 12{,,,}{1,2,,}m i i i k ⊂L L ,1212()()()()m mP A A A P A P A P A i i i i i i =I I L L (全概率公式 total probability)()(|)()i i P B A P B A P A =IUsing the theorem of total probability, we have1()(|)(|)()(|)i i i k j jj P B P A B P B A P B P A B ==∑ 1,2,,i k =L1、 random variable definition2、 Distribution functionNote The distribution function ()F X is defined on real numbers, not on sample space 、3、 PropertiesThe distribution function ()F x of a random variable X has the following properties:3、2 Discrete Random Variables 离散型随机变量geometric distribution (几何分布)Binomial distribution(二项分布)poisson distribution(泊松分布)Expectation (mean) 数学期望2.Variance 方差standard deviation (标准差)probability density function概率密度函数5、 Mean(均值)6、 variance 方差、4、2 Uniform Distribution 均匀分布The uniform distribution, with the parameters a a nd b , has probability density function1 for ,()0 elsewhere,a xb f x b a ⎧<<⎪=-⎨⎪⎩4、5 Exponential Distribution 指数分布4、3 Normal Distribution 正态分布1、Definition4、4 Normal Approximation to the Binomial Distribution(二项分布)4、7 Chebyshev’s Theorem(切比雪夫定理)Joint probability distribution(联合分布)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 values specified for that variable、5.2C onditional distribution 条件分布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(,)(|)()X Y p x y p x y p y = for all x provided ()0Y p y ≠、 5.3 S tatistical independent 随机变量的独立性interdependence of X and Y we want to examine 、number of random phenomenon 、 And the average of large number of random variables are also steadiness 、 These results are the law of large numbers 、population (总体)sample (样本、子样)中位数It is customary to write )(X E as X μ and )(X D as 2X σ、Here, ()E X μ= is called the expectation of the mean 、均值的期望 n X σσ= is called the standard error of the mean 、 均值的标准差 7、1 Point Estimate 点估计Unbiased estimator(无偏估计量)minimum variance unbiased estimator(最小方差无偏估计量)3、Method of Moments 矩估计的方法confidence interval----- 置信区间lower confidence limits-----置信下限upper confidence limits----- 置信上限degree of confidence----置信度2.极大似然函数likelihood function显著性水平Two Types of Errors。

第1 章随机事件及其概率在一个试验下，不管事件有多少个，总可以从其中找出这样一组事件，它具有如下性质：①每进行一次试验，必须发生且只能发生这一组中的一个事件；②任何事件，都是由这一组中的部分事件组成的。

这样一组事件中的每一个事件称为基本事件，用来表示。

基本事件的全体，称为试验的样本空间，用Ω表示。

一个事件就是由Ω中的部分点（基本事件）组成的集合。

通常用大写字母A，B，C，…表示事件，它们是Ω的子集。

Ω为必然事件，Ø 为不可能事件。

不可能事件（Ø）的概率为零，而概率为零的事件不一定是不可能事件；同理，必然事件（Ω）的概率为 1，而概率为 1 的事件也不一定是必然事件。

1°Ω={1,2 n}，12°P(1) =P(2) = P(n) =n。

设任一事件A ，它是由1,2 m组成的，则有P(A)= {(1) (2) (m)}= P(1) +P(2) + +P(m)=m=A所包含的基本事件数n 基本事件总数第二章随机变量及其分布设随机变量X 的分布律为k-P( X =k ) = e ，> 0 ，k = 0,1,2 ，k!则称随机变量X 服从参数为的泊松分布，记为X ~ () 或者P()。

泊松分布为二项分布的极限分布（np=λ，n→∞）。

e-x , x≥0,f (x) =0, x < 0 ,其中> 0 ，则称随机变量 X 服从参数为的指数分布。

X 的分布函数为1 -e-x, x≥0,F (x) =0,x<0。

记住积分公式：+∞⎰x n e -x dx =n!正态分布设随机变量 X 的密度函数为 21-( x -) 2- ∞ < x < +∞f (x ) =e 2 ， ， 2其中、> 0 为常数，则称随机变量 X 服从参数为、的正态分布或高斯（Gauss ）分布，记为 X ~ N (,2) 。

f (x ) 具有如下性质：1° f (x ) 的图形是关于 x = 对称的；2° 当 x = 时， f ()= 1 为最大值；若 X ~ N (,2) ，则 X 2 的分布函数为1 x e- ( t - ) 2F (x ) =⎰- 2 2dt2∞参数= 0 、= 1时的正态分布称为标准正态分布，记为X ~ N (0,1) ，1 其-密x 2度函数记为 (x ) = e 22 ， - ∞ < x < +∞ ，分布函数为21x - t Φ(x ) = 2⎰ e 2dt 。