概率论与数理统计英文版总结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. Assume that 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 multiplication theorem 乘法定理 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 subset12{,,,}{1,2,,}m i i i k ⊂L L ,1212()()()()m m P 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)（贝叶斯公式Bayes ’ formula.）()(|)()i i P B A P B A P A =IUsing the theorem of total probability, we have1()(|)(|)()(|)i i i kjjj 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 function1for ,()0 elsewhere,a x b f x b a⎧<<⎪=-⎨⎪⎩4.5 Exponential Distribution 指数分布4.3 Normal Distribution 正态分布4.4 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. 5.2 C 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 ≠.5.3S tatistical independent 随机变量的独立性5.4 Covariance and Correlation 协方差和相关系数We now define two related quantities whose role in characterizing the interdependence of X and Y we want to examine.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.population (总体)sample （样本、子样）中位数Sample Distributions 抽样分布1．sampling distribution of the mean 均值的抽样分布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 functionmaximum likelihood estimate（最大似然估计）8.1 Statistical Hypotheses（统计假设）显著性水平Two Types of Errors。