《概率论与数理统计英文版总结(总14页)》由会员分享，可在线阅读，更多相关《概率论与数理统计英文版总结(总14页)（14页珍藏版）》请在金锄头文库上搜索。
1、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 itself, is certainly an event, which is called a certain event, means that it always occurs in the experiment.
2、 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 trails is denoted by , and if the sequence of relative frequencies obtained for larger and larger va
3、lue 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
4、 occurs 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 事件的独立性 Two events and are said to be independent if Or Two events and are independent if and only if .Conditional Probability 条
5、件概率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
6、 2.6.1. If the events constitute a partition of the sample space S such that for than for any event of , (2.6.2)（贝叶斯公式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
7、 probability, 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
8、random 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 ,
9、 then 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 离散型随机变量Definition 3.2.1 A random variable is called a discrete random variable, if it takes values from a finite set or,
10、 a set whose elements can be written as a sequence geometric distribution （几何分布） X 1234kPpq1pq2pq3pqk1pBinomial distribution（二项分布）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
11、 the 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) 数学期望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 （标准差）Definition 3.3.2 Let be a discrete random variable, having expectation . Then the variance of , denote by is defined as the expectation of
13、 the 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 概率密度函数Definition 4.1.1 A function f(x) defined on is called a probability density function （概率密度函数）if:(i) ;(ii) f(x) is intergrable (可积的) on and .Defin
14、ition 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 probability density function f(x). Then the mean (or expectation) of X is defined by, (4.1.3)provided the integral converges ab
