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 variable (r,v.)
2.3 Probability
2.4 R.V,and probability density function
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
(,) ( ) ( )XYf 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
211( ) e x p { ( ) }
22
x
fx


2) Sampling distribution for a function of a sample
example:
3) Chi-squared distribution,
2
1
2
,,~ (,)
t h e n ~ (,/ )
nX X N
X N n


1
2 2 2 2
1 ( )
,,~ ( 0,1 )
t h e n ~
n
nn
X X N
XX
4) t-distribution
2
1
2
2
2 1
( 1 )
,,~ (,)
t he n ~ (,/ ) a nd ~ ( 0,1 )
()
~ w he r e
1
n
n
i
i
n
X X N
X
X N n N
n
XX
X
t t S
S n
n



5) F-distribution
2
2
( 1,
22
11
22
22
1)2
11
2
,,~ (,),,,~ (,)
( ) ( )
,
11
t he n
~
m X X n Y Y
mn
ii
ii
XY
X
X
mn
Y
Y
X X N Y Y N
X X Y Y
SS
mn
S
F
S






2.9.1 统计推断的两大部分,
2.9.2 Parameter estimation
1) Point estimation
2) Interval estimation
p a r a m e t e r e s t i m a t i n gS t a t i s t i c a l I n f e r e n c e
h y p o t h e s i s t e s t i n g
11{ (,,) (,,) } 1nnP L X X U X X
2.9.3 Properties of a point estimator
1) Linearity (sample mean)
2) Unbiasedness
3) Efficiency
4) BLUE
5) Consistency
2.9.4 Hypothesis testing
1) Acceptance region
2) Critical region=rejection region
3) Level of significance
4) Probability of committing a type I error
5) Power of test
Test of significance approach
1) t-test
2) Chi-squared test
3) F-test