厦门大学：《高级经济计量学》讲义 第二章 概率与统计回顾

Chapter 2 概率与统计回顾

Chapter 2 概率与统计回顾

2.1 1 Statistical or random experiment 2) Sample space or population Sample point, event 2. 2 Stochastic or random variabler. v) 2.3 Probability 2. 4R.V. and probability density function

2.1 1) Statistical or random experiment 2) Sample space or population Sample point, event 2.2 Stochastic or random variable (r. v.) 2.3 Probability 2.4 R.V. and probability density function

2.5 Multiple random variable Statistically independence f(x,y)=g(h(y) 2.6 Numerical characteristics of ar v 1)expected value(mean) 2)variance and rth moment 3covariance and correlation 4)skewness and kurtosis

2.5 Multiple random variable Statistically independence 2.6 Numerical characteristics of a r. v. 1) expected value (mean) 2) variance and rth moment 3) covariance and correlation 4) skewness and kurtosis ( , ) ( ) ( ) X Y f x y g x h y =

2.7 Sample and functions of a sample Sample moment 2.8 Some important probability distributions 1)normal distribution ()=exp(- 32 2丌 2

2.7 Sample and functions of a sample Sample moment 2.8 Some important probability distributions 1) normal distribution 1 1 2 ( ) exp{ ( ) } 2 2 x f x     − = −

2)Sampling distribution for a function of a sample example X, .,,X,N(u,o) then XN(,o/n) 3Chi-squared distribution 1…,Xn~N(0,1) then X"2+…+X

2) Sampling distribution for a function of a sample example: 3) Chi-squared distribution: 2 1 2 , , ~ ( , ) then ~ ( , / ) X X N n X N n     1 2 2 2 2 1 ( ) , , ~ (0,1) then ~ n n n X X N   = + + X X

4)t-distribution 1,…,Xn~N(A,a2) then X N(u,o /n) and X-H~N(0,1) X-p ∑(X-X) where S2=is

4) t-distribution 2 1 2 2 2 1 ( 1) , , ~ ( , ) then ~ ( , / ) and ~ (0,1) ( ) ~ where 1 n n i i n X X N X X N n N n X X X t t S S n n        = − − − − = = − 

5)F-distribution X12…,Xm~N(x2Ox2),H1…,Yn~N(y2Oy2) ∑(X1-x)2 (Y-Y) 2 2 X Y en X 2 2 (m-1,n-1) Y

5) F-distribution 2 2 ( 1, 2 2 1 1 2 2 2 2 2 1) 1 1 2 , , ~ ( , ), , , ~ ( , ) ( ) ( ) , 1 1 then ~ m X X n Y Y m n i i i i X Y X X m n Y Y X X N Y Y N X X Y Y S S m n S F S       = − − = − − = = − −  

2.9 Statistical Inference

291统计推断的两大部分 Statistical Inference, parameter estimating hypothesis testing 2.9.2 Parameter estimation 1)Point estimation 2)Interval estimation PL(Xx,…,)≤≤U(X,…,Xn)}=1-a

2.9.1 统计推断的两大部分: 2.9.2 Parameter estimation 1) Point estimation 2) Interval estimation parameter estimating Statistical Inference hypothesis testing    1 1 { ( , , ) ( , , )} 1 P L X X U X X n n   = −  

2.9. 3 Properties of a point estimator 1)Linearity(sample mean 2)Unbiasedness 3)Efficiency 4) BLUE 5)Consistency

2.9.3 Properties of a point estimator 1) Linearity (sample mean) 2) Unbiasedness 3) Efficiency 4) BLUE 5) Consistency