概率论与数理统计英文版总结.doc

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.
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 value 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 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 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 probability,

Using the theorem of total probability, we have

1. random variable definition
Definition 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 function
Definition 3.1.2 Let be a 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. Properties
The distribution function of a random variable has the following properties:
(1) is non-decreasing.
In fact, if , 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, a set whose elements can be written as a sequence
geometric distribution （几何分布）
X
1
2
3
4
…
k
…
P
p
q1p
q2p
q3p

qk－1p
…
Binomial 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 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 .
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)
2．Variance 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 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 .

Definition 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 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 deviation.
We easily get
. (4.1.5)
.
4.2 Uniform Distribution 均匀分布
The uniform distribution, with the parameters a and b, has probability density function

4.5 Exponential Distribution 指数分布Definition 4.5.1 A continuous variable X has an exponential distribution with parameter , if its density function is given 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 正态分布
1. Definition
The equation of the normal probability density, whose graph is shown in Figure 4.3.1, is

4.4 Normal Approximation to the Binomial Distribution（二项分布）
, n is large (n>30), p is close to 0.50,

4.7 Chebyshev’s Theorem（切比雪夫定理）
Theorem 4.7.1 If a probability distribution has mean μ and standard deviation σ, the probability of getting a value which deviates from μ by at least kσ is at most . Symbolically ,
.
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 Conditional 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
for all x provided .
5.3 Statistical independent 随机变量的独立性
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 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.
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 to be uncorrelated iff .

5.5 Law of Large Numbers and Central Limit Theorem 中心极限定理
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 5.5.1 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)
(频率具有稳定性)
Theorem 5.5.3 If is independent, with

then .

population (总体)
Definition 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 （样本、子样）
Definition 6.2.2 A sample is a subset of the population from which people can draw conclusions about the whole.

sampling(抽样)
taking a sample: The process of performing an experiment to obtain a sample from the population is called sampling.
中位数
Definition 6.2.4 If a random sample has the order statistics , then
(i) The Sample Median is

(ii) The Sample Range is
.

Sample Distributions 抽样分布
1．sampling distribution of the mean 均值的抽样分布
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 the mean. 均值的标准差
7.1 Point Estimate 点估计
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 a point estimate of . The point estimator is also often written as .
Unbiased estimator(无偏估计量)
Definition 7.1.2. Suppose is an estimator of a parameter . Then is unbiased if and only if

minimum variance unbiased estimator（最小方差无偏估计量）
Definition 7.1.3 Let be an unbiased estimator of . If for any which is 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 矩估计的方法
Definition 7.1.4
Suppose constitute a random sample from the population X that has k unknown parameters . Also, the population 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 random 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
Definition 7.5.1 A random sample has the observed values from a population with an unknown parameter . Then the likelihood function for this sample is

in which is defined in (7.5.1).

maximum likelihood estimate （最大似然估计）
Definition 7.5.2 If there is a value such that

for all , then is called a maximum likelihood estimate of .

8.1 Statistical Hypotheses（统计假设）
Definition 8.1.1 A statistical hypothesis is an assertion or conjecture concerning one or more population.

Definition 8.1.2 A statistical procedure or decision rule that leads to establishing the truth or falsity of a hypothesis is called a statistical test.

显著性水平
Definition 8.2.1 Which is called significant level, describes how far the sample mean is far from the population mean.

Two Types of Errors
Definition 8.2.3 If it happens that the hypothesis being tested is actually true, and if from the sample we reach the conclusion that it is false, we say that a type I error has been committed.

Definition 8.2.4 If it happens that hypothesis being tested is actually false, and if from the sample we reach the conclusion that it is true, we say that a type II error has been committed.