第 2章 概率分布
?Normal Distribution
?Other Continuous Distributions
本章概要
?The Normal Distribution
?The Standard Normal Distribution
?Assessing the Normality Assumption
?The Exponential Distribution
?Sampling Distribution of the Mean
?Sampling Distribution of the Proportion
?Sampling From Finite Populations
Continuous Probability Distributions
连续的概率分布
?Continuous Random Variable:
Values from Interval of Numbers
Continuous Probability Distribution:
Distribution of a Continuous Variable
?Most Important Continuous Probability
Distribution,the Normal Distribution
The Normal Distribution
正态分布
? Bell Shaped
? Symmetrical
? Mean,Median and
Mode are Equal
? Middle Spread?
Equals 1.33 ??
? Random Variable has
Infinite Range
Mean
Median
Mode
X
f(X)
?
概率密度函数
(Probability Density Function,p.d.f,)
f(X) = frequency of random variable X
? = 3.14159; e = 2.71828
? = population standard deviation
X = value of random variable (-? < X < ?)
? = population mean (总体均数 )
f(X) = 1 e (-1/2) ((X-????)
2
??2
多个正态分布的比较
Varying the Parameters ? and ?,we obtain
Different Normal Distributions.
There are an
Infinite
Number
Normal Distribution,Finding Probabilities
正态分布曲线下面积的几何意义
Probability is the
area under the
curve ! 累积概率
c d X
f(X)
P c X d( )?? ? ?
每个正态分布对应一张正态概率表
Which Table? 请用标准正态分布！
Infinitely Many Normal Distributions Means
Infinitely Many Tables to Look Up!
Each distribution has
its own table?
Z Z
Z = 0.12
Z,00,01
0.0,0000,0040,0080
.0398,0438
0.2,0793,0832,0871
0.3,0179,0217,0255
The Standardized Normal Distribution
标准正态分布
.0478.02
0.1,0478
Standardized Normal Probability
Table (Portion) ? = 0 and ?? = 1
Probabilities
Shaded Area
Exaggerated
Z? = 0
?Z = 1
.12
标准化的应用（ 1）
Normal
Distribution
Standardized Normal
Distribution
X? = 5
? = 10
6.2
120
10
526,.XZ ?????
?
?
Shaded Area Exaggerated
0
? = 1
-.21 Z.21
标准化应用（ 2）
P(2.9 < X < 7.1) =,1664
Normal
Distribution
.1664
.0832.0832
Standardized
Normal Distribution
Shaded Area Exaggerated
5
? = 10
2.9 7.1 X
21
10
592,.xz ??????
?
?
2110 517,.xz ????? ? ?
Z? = 0
? = 1
.30
Example,P(X ? 8) =,3821
Normal
Distribution
Standardized Normal
Distribution
.1179
.5000
.3821
Shaded Area Exaggerated
.
X? = 5
? = 10
8
30
10
58
.
x
z ?
?
?
?
?
?
?
Z,00 0.2
0.0,0000,0040,0080
0.1,0398,0438,0478
0.2,0793,0832,0871
.1179,1255
Z? = 0
? = 1
.31
Finding Z Values for Known Probabilities
已知概率求 Z值
.1217,01
0.3
Standardized Normal
Probability Table (Portion)
What Is Z Given
P(Z) = 0.1217?
Shaded Area
Exaggerated
.1217
Z? = 0
? = 1
.31X? = 5
? = 10
Finding X Values for Known Probabilities
已知概率求 X值
Normal Distribution Standardized Normal Distribution
.1217,1217
Shaded Area Exaggerated
X 8.1??????Z??= 5 + (0.31)(10) =
Assessing Normality
正态性评价 ---借助标化公式
Compare Data Characteristics
to Properties of Normal
Distribution
? Put Data into Ordered Array
? Find Corresponding Standard
Normal Quantile Values
? Plot Pairs of Points
? Assess by Line Shape
Normal Probability Plot
for Normal Distribution
Look for Straight Line!
30
60
90
-2 -1 0 1 2
Z
X
Normal Probability Plots
Left-Skewed
左偏态
Right-Skewed
右偏态
Rectangular U-Shaped
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
Exponential Distributions
指数分布
P arrival time < X( ) ? 1 - e
e = the mathematical constant 2.71828
-? x
???= the population mean of arrivals???
X = any value of the continuous random
variable
e.g,Drivers Arriving at a Toll Bridge
Customers Arriving at an ATM Machine
Describes time or distance
between events
? Used for queues
Density function
Parameters
指数分布密度函数及参数
f(x) = 1? e -x??
???????????????
f(X)
X
? = 0.5
? = 2.0
Estimation of Parameter
参数估计
?Sample Statistic Estimates Population Parameter
e.g,X = 50 estimates Population Mean,?
?Problems, Many samples provide many estimates of the
Population Parameter.
Determining adequate sample size,large sample give better estimates,
Large samples more costly.
How good is the estimate?
?Approach to Solution,Theoretical Basis is Sampling
Distribution （抽样分布）,
_
Sampling Distributions
抽样分布
?Theoretical Probability Distribution
? Random Variable is Sample Statistic:
Sample Mean,Sample Proportion (样本均数、样本比例 )
? Results from taking All Possible Samples of
the Same Size
?Comparing Size of Population and Size of Sampling Distribution
Population Size = 100
Size of Samples = 10
Sampling Distribution Size =
(Sampling Without Replacement 不返还抽样 )
1.73?10 13
Population size,N = 4
Random variable,X,
is Age of individuals
Values of X,18,20,22,24
measured in years
1984-1994 T/Maker Co.
抽样分布的拓展
A
B C
D
Suppose there is a population...
? ?
236.2
21
4
24222018
1
2
1
?
?
?
?
???
?
?
?
?
?
?
N
X
N
X
N
i
i
N
i
i
?
?
?
Population Characteristics
总体特征
Summary Measure Population Distribution
.3
.2
.1
0
A B C D
(18) (20) (22) (24)
Uniform Distribution
P(X)
X
1
st
2
nd
Ob serv atio n
Obs 18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
16 Samples
Samples Taken with Replacement
有返还抽样
16 Sample Means
All Possible Samples of Size n = 2
样本含量 =2 的所有返还抽样
1st 2 n d O b s e r v a t i o n
O b s 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
1st 2 n d O b s e r v a t i o n
O b s 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
1st 2 n d O b s e r v a t i o n
O b s 18 2 0 22 2 4
18 18 19 20 21
2 0 19 20 21 22
22 20 21 22 23
2 4 21 22 23 24
1st 2 n d O b s e r v a t i o n
O b s 18 2 0 22 2 4
18 18 19 20 21
2 0 19 20 21 22
22 20 21 22 23
2 4 21 22 23 24
18 19 20 21 22 23 240
.1
.2
.3
P(X)
X
Sample Means
Distribution
16 Sample Means
Sampling Distribution of All Sample Means
所有样本均数的抽样分布
# in sample = 2,# in Sampling Distribution = 16
_
21
16
241919181
?
????
?
?
? ?
?
N
X
N
i
i
x?
? ?
? ? ? ? ? ?
581
16
212421192118
222
1
2
.
N
X
N
i
xi
x
?
??????
?
? ?
?
?
?
?
?
抽样分布的描述
18 19 20 21 22 23 240
.1
.2
.3
P(X)
X
样本均数的分布
n = 2
Comparing the Population with its
Sampling Distribution
A B C D
(18) (20) (22) (24)
0
.1
.2
.3
总 体
N = 4
???= 21,??= 2.236
P(X)
X
21?x? 581,x ??
_
? Population Mean Equal to
Sampling Mean
? The Standard Error (standard deviation) of
the Sampling distribution is Less than
Population Standard Deviation
? Formula (sampling with replacement):
抽样分布参数的特征
?? ?x
As n increase,decrease.??x = ?? ??xn __
Unbiasedness （无偏性）
? Mean of sampling distribution equals population
mean
Efficiency （有效性）
? Sample mean comes closer to population mean than
any other unbiased estimator
Consistency （一致性）
? As sample size increases,variation of sample mean
from population mean decreases
均数 (估计 )的特征
?
无偏性
Biased 有偏的Unbiased
P(X)
X
?
有效性
Sampling
Distribution
of Median Sampling
Distribution of
Mean
X
P(X)
?
Larger
sample size
Smaller
sample size
一致性
X
P(X)
A
B
?
= 5 0
? ?= 1 0
X? = 5 0
? ?= 1 0
X
?
X
= 5 0- X?
X
= 5 0- Xn =16
??X = 2.5
n = 4
??X = 5
正态总体情形
Central Tendency
Variation
Sampling with
Replacement
Population Distribution
Sampling Distributions
??x ??=
?? ??x =
n
_
_
XX
Central Limit Theorem
中心极限定理
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost Normal
regardless of
shape of
population
nx
?? ?
?? ?x
n =30
??X = 1.8n = 4
??X = 5
非正态总体情形
Central Tendency
集中趋势
Variation
Sampling with
Replacement
Population Distribution
Sampling Distributions
? = 50
? = 10
X
X50??
X
抽样分布的应用
Sampling
Distribution
Standardized
Normal Distribution
.1915
50
252
887,
/
.
n/
XZ ??????
?
?
4.X ??
7.8 8 8.2 ???? = 0 Z
? = 1
.3830
.1915
50
252
828,
/
.
n/
XZ ?????
?
?
? Categorical variable (e.g.,gender) 分类变量
? % population having a characteristic
? If two outcomes,binomial distribution
?Possess or don’t possess characteristic
? Sample proportion (ps)
s i z es a m pl e
s uc c e s s e s of nu m be r
n
X
P s ??
Population Proportions
总体比例
Approximated by normal
distribution
? np ? 5
? n?1 - p)?? 5
Mean
Standard error
pP ??
? ?
n
pp
P
??? 1?
比例的抽样分布
p = population proportion
Sampling Distribution
?? P(ps).3
.2
.1
0
0, 2,4,6 8 1 ps
正态近似
Standardizing Sampling Distribution of Proportion
标准化比例的抽样分布
Sampling
Distribution
Standardized
Normal Distribution
Z pp s? s - ? p?
p
= p -
n
)p(p ?1
ps Z???? = 0
?p
????????p
? = 1
比例抽样分布的应用
51
5
??
?
)p(n
np
Sampling
Distribution Normal DistributionStandardized
?
Z? ps -
-
p =,43 -,40 =,87
n
)p(p ?1
2 0 0
40140 ).(,??
?p =,0346
ps
? = 1
? = 0,87 Z
..3078
?p =,40,43
? Modify Standard Error if Sample Size (n) is
Large Relative to Population Size (N)
n >,05N (or n/N >,05)
? Use Finite Population Correction Factor
(fpc) （校正因子）
? Standard errors if n/N >,05:
1?
???
N
nN
nx
?? ? ? ? ?
? ?1
1
?
?????
N
nN
n
pp
P?
有限总体的抽样
本章 小结
?Discussed The Normal Distribution
?Described The Standard Normal Distribution
?Assessed the Normality Assumption
?Defined The Exponential Distribution
?Discussed Sampling Distribution of the Mean
?Described Sampling Distribution of the Proportion
?Defined Sampling From Finite Populations
?Normal Distribution
?Other Continuous Distributions
本章概要
?The Normal Distribution
?The Standard Normal Distribution
?Assessing the Normality Assumption
?The Exponential Distribution
?Sampling Distribution of the Mean
?Sampling Distribution of the Proportion
?Sampling From Finite Populations
Continuous Probability Distributions
连续的概率分布
?Continuous Random Variable:
Values from Interval of Numbers
Continuous Probability Distribution:
Distribution of a Continuous Variable
?Most Important Continuous Probability
Distribution,the Normal Distribution
The Normal Distribution
正态分布
? Bell Shaped
? Symmetrical
? Mean,Median and
Mode are Equal
? Middle Spread?
Equals 1.33 ??
? Random Variable has
Infinite Range
Mean
Median
Mode
X
f(X)
?
概率密度函数
(Probability Density Function,p.d.f,)
f(X) = frequency of random variable X
? = 3.14159; e = 2.71828
? = population standard deviation
X = value of random variable (-? < X < ?)
? = population mean (总体均数 )
f(X) = 1 e (-1/2) ((X-????)
2
??2
多个正态分布的比较
Varying the Parameters ? and ?,we obtain
Different Normal Distributions.
There are an
Infinite
Number
Normal Distribution,Finding Probabilities
正态分布曲线下面积的几何意义
Probability is the
area under the
curve ! 累积概率
c d X
f(X)
P c X d( )?? ? ?
每个正态分布对应一张正态概率表
Which Table? 请用标准正态分布！
Infinitely Many Normal Distributions Means
Infinitely Many Tables to Look Up!
Each distribution has
its own table?
Z Z
Z = 0.12
Z,00,01
0.0,0000,0040,0080
.0398,0438
0.2,0793,0832,0871
0.3,0179,0217,0255
The Standardized Normal Distribution
标准正态分布
.0478.02
0.1,0478
Standardized Normal Probability
Table (Portion) ? = 0 and ?? = 1
Probabilities
Shaded Area
Exaggerated
Z? = 0
?Z = 1
.12
标准化的应用（ 1）
Normal
Distribution
Standardized Normal
Distribution
X? = 5
? = 10
6.2
120
10
526,.XZ ?????
?
?
Shaded Area Exaggerated
0
? = 1
-.21 Z.21
标准化应用（ 2）
P(2.9 < X < 7.1) =,1664
Normal
Distribution
.1664
.0832.0832
Standardized
Normal Distribution
Shaded Area Exaggerated
5
? = 10
2.9 7.1 X
21
10
592,.xz ??????
?
?
2110 517,.xz ????? ? ?
Z? = 0
? = 1
.30
Example,P(X ? 8) =,3821
Normal
Distribution
Standardized Normal
Distribution
.1179
.5000
.3821
Shaded Area Exaggerated
.
X? = 5
? = 10
8
30
10
58
.
x
z ?
?
?
?
?
?
?
Z,00 0.2
0.0,0000,0040,0080
0.1,0398,0438,0478
0.2,0793,0832,0871
.1179,1255
Z? = 0
? = 1
.31
Finding Z Values for Known Probabilities
已知概率求 Z值
.1217,01
0.3
Standardized Normal
Probability Table (Portion)
What Is Z Given
P(Z) = 0.1217?
Shaded Area
Exaggerated
.1217
Z? = 0
? = 1
.31X? = 5
? = 10
Finding X Values for Known Probabilities
已知概率求 X值
Normal Distribution Standardized Normal Distribution
.1217,1217
Shaded Area Exaggerated
X 8.1??????Z??= 5 + (0.31)(10) =
Assessing Normality
正态性评价 ---借助标化公式
Compare Data Characteristics
to Properties of Normal
Distribution
? Put Data into Ordered Array
? Find Corresponding Standard
Normal Quantile Values
? Plot Pairs of Points
? Assess by Line Shape
Normal Probability Plot
for Normal Distribution
Look for Straight Line!
30
60
90
-2 -1 0 1 2
Z
X
Normal Probability Plots
Left-Skewed
左偏态
Right-Skewed
右偏态
Rectangular U-Shaped
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
Exponential Distributions
指数分布
P arrival time < X( ) ? 1 - e
e = the mathematical constant 2.71828
-? x
???= the population mean of arrivals???
X = any value of the continuous random
variable
e.g,Drivers Arriving at a Toll Bridge
Customers Arriving at an ATM Machine
Describes time or distance
between events
? Used for queues
Density function
Parameters
指数分布密度函数及参数
f(x) = 1? e -x??
???????????????
f(X)
X
? = 0.5
? = 2.0
Estimation of Parameter
参数估计
?Sample Statistic Estimates Population Parameter
e.g,X = 50 estimates Population Mean,?
?Problems, Many samples provide many estimates of the
Population Parameter.
Determining adequate sample size,large sample give better estimates,
Large samples more costly.
How good is the estimate?
?Approach to Solution,Theoretical Basis is Sampling
Distribution （抽样分布）,
_
Sampling Distributions
抽样分布
?Theoretical Probability Distribution
? Random Variable is Sample Statistic:
Sample Mean,Sample Proportion (样本均数、样本比例 )
? Results from taking All Possible Samples of
the Same Size
?Comparing Size of Population and Size of Sampling Distribution
Population Size = 100
Size of Samples = 10
Sampling Distribution Size =
(Sampling Without Replacement 不返还抽样 )
1.73?10 13
Population size,N = 4
Random variable,X,
is Age of individuals
Values of X,18,20,22,24
measured in years
1984-1994 T/Maker Co.
抽样分布的拓展
A
B C
D
Suppose there is a population...
? ?
236.2
21
4
24222018
1
2
1
?
?
?
?
???
?
?
?
?
?
?
N
X
N
X
N
i
i
N
i
i
?
?
?
Population Characteristics
总体特征
Summary Measure Population Distribution
.3
.2
.1
0
A B C D
(18) (20) (22) (24)
Uniform Distribution
P(X)
X
1
st
2
nd
Ob serv atio n
Obs 18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
16 Samples
Samples Taken with Replacement
有返还抽样
16 Sample Means
All Possible Samples of Size n = 2
样本含量 =2 的所有返还抽样
1st 2 n d O b s e r v a t i o n
O b s 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
1st 2 n d O b s e r v a t i o n
O b s 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
1st 2 n d O b s e r v a t i o n
O b s 18 2 0 22 2 4
18 18 19 20 21
2 0 19 20 21 22
22 20 21 22 23
2 4 21 22 23 24
1st 2 n d O b s e r v a t i o n
O b s 18 2 0 22 2 4
18 18 19 20 21
2 0 19 20 21 22
22 20 21 22 23
2 4 21 22 23 24
18 19 20 21 22 23 240
.1
.2
.3
P(X)
X
Sample Means
Distribution
16 Sample Means
Sampling Distribution of All Sample Means
所有样本均数的抽样分布
# in sample = 2,# in Sampling Distribution = 16
_
21
16
241919181
?
????
?
?
? ?
?
N
X
N
i
i
x?
? ?
? ? ? ? ? ?
581
16
212421192118
222
1
2
.
N
X
N
i
xi
x
?
??????
?
? ?
?
?
?
?
?
抽样分布的描述
18 19 20 21 22 23 240
.1
.2
.3
P(X)
X
样本均数的分布
n = 2
Comparing the Population with its
Sampling Distribution
A B C D
(18) (20) (22) (24)
0
.1
.2
.3
总 体
N = 4
???= 21,??= 2.236
P(X)
X
21?x? 581,x ??
_
? Population Mean Equal to
Sampling Mean
? The Standard Error (standard deviation) of
the Sampling distribution is Less than
Population Standard Deviation
? Formula (sampling with replacement):
抽样分布参数的特征
?? ?x
As n increase,decrease.??x = ?? ??xn __
Unbiasedness （无偏性）
? Mean of sampling distribution equals population
mean
Efficiency （有效性）
? Sample mean comes closer to population mean than
any other unbiased estimator
Consistency （一致性）
? As sample size increases,variation of sample mean
from population mean decreases
均数 (估计 )的特征
?
无偏性
Biased 有偏的Unbiased
P(X)
X
?
有效性
Sampling
Distribution
of Median Sampling
Distribution of
Mean
X
P(X)
?
Larger
sample size
Smaller
sample size
一致性
X
P(X)
A
B
?
= 5 0
? ?= 1 0
X? = 5 0
? ?= 1 0
X
?
X
= 5 0- X?
X
= 5 0- Xn =16
??X = 2.5
n = 4
??X = 5
正态总体情形
Central Tendency
Variation
Sampling with
Replacement
Population Distribution
Sampling Distributions
??x ??=
?? ??x =
n
_
_
XX
Central Limit Theorem
中心极限定理
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost Normal
regardless of
shape of
population
nx
?? ?
?? ?x
n =30
??X = 1.8n = 4
??X = 5
非正态总体情形
Central Tendency
集中趋势
Variation
Sampling with
Replacement
Population Distribution
Sampling Distributions
? = 50
? = 10
X
X50??
X
抽样分布的应用
Sampling
Distribution
Standardized
Normal Distribution
.1915
50
252
887,
/
.
n/
XZ ??????
?
?
4.X ??
7.8 8 8.2 ???? = 0 Z
? = 1
.3830
.1915
50
252
828,
/
.
n/
XZ ?????
?
?
? Categorical variable (e.g.,gender) 分类变量
? % population having a characteristic
? If two outcomes,binomial distribution
?Possess or don’t possess characteristic
? Sample proportion (ps)
s i z es a m pl e
s uc c e s s e s of nu m be r
n
X
P s ??
Population Proportions
总体比例
Approximated by normal
distribution
? np ? 5
? n?1 - p)?? 5
Mean
Standard error
pP ??
? ?
n
pp
P
??? 1?
比例的抽样分布
p = population proportion
Sampling Distribution
?? P(ps).3
.2
.1
0
0, 2,4,6 8 1 ps
正态近似
Standardizing Sampling Distribution of Proportion
标准化比例的抽样分布
Sampling
Distribution
Standardized
Normal Distribution
Z pp s? s - ? p?
p
= p -
n
)p(p ?1
ps Z???? = 0
?p
????????p
? = 1
比例抽样分布的应用
51
5
??
?
)p(n
np
Sampling
Distribution Normal DistributionStandardized
?
Z? ps -
-
p =,43 -,40 =,87
n
)p(p ?1
2 0 0
40140 ).(,??
?p =,0346
ps
? = 1
? = 0,87 Z
..3078
?p =,40,43
? Modify Standard Error if Sample Size (n) is
Large Relative to Population Size (N)
n >,05N (or n/N >,05)
? Use Finite Population Correction Factor
(fpc) （校正因子）
? Standard errors if n/N >,05:
1?
???
N
nN
nx
?? ? ? ? ?
? ?1
1
?
?????
N
nN
n
pp
P?
有限总体的抽样
本章 小结
?Discussed The Normal Distribution
?Described The Standard Normal Distribution
?Assessed the Normality Assumption
?Defined The Exponential Distribution
?Discussed Sampling Distribution of the Mean
?Described Sampling Distribution of the Proportion
?Defined Sampling From Finite Populations
