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# 統計學文本分析

## 第 4 章 研究結果分析

### 4.2.2 統計學文本分析

Probability Distribution , Mean , and Variance for a Poisson Random Variable

( ) ( 0,1, 2...)

!

xe

p x x

x λ λ

= =

2

µ λ

σ λ

=

=

where λ= Mean number of events during given unit of time , area , volumn , etc.

e= 2.71828

Chapter 4 Random Variables amd Probability Distribution

4.4 The Poisson Distribution (Optional)

Poisson 分佈式離散型機率分佈的 其中一種，其概念為：若隨機變數 X 是在已知時間間隔或某一指定區 域中所發生的出象數，則產生此隨 機變數數值的實驗稱為 Poisson experiments。實驗期間發生的出象 數目 X 稱為 Poisson random variable，其機率分佈稱為 Poisson distribution。

p.218

Students with knowledge of calculus should note that the probability that x assumes a value in the interval a< < x b is ( ) b ( )

P a< <x b =

### ∫

a f x dx，assuming the integral exists . Similar to the requirement for a discrete probability distribution , we require f x( )≥ and 0 f x dx( ) 1

−∞ =

### ∫

.

Chapter 4 Random Variables amd Probability Distribution

4.6 The Uniform Distribution (Optional)

( )

P a< <x b 為在區間

a< < ， ( )x b f x 曲線下的面積，

p.224

Probability Distribution for a Uniform Random Variable x

Chapter 4 Random Variables amd

Probability Distribution p.225

Probabilty density function：

( ) 1

f x c x d

d c

= ≤ ≤

− Mean：

2 c d µ= + Standard deviation：

12 d c

σ =

( ) ( ) /( ),

P a< <x b = ba dc c≤ < ≤a b d

The students with knowledge of calculus should note that

( ) b ( )

a

P a< <x b =

### ∫

f x dx ( ) 1/( )

b b

a f x dx a d c dx

=

=

### ∫

− (b a) /(d c)

= − −

4.6 The Uniform Distribution (Optional)

Probability Distribution for a Normal Random Variable x

Probabilty density function：

(1/ 2)[( ) / ]2

( ) 1 2

f x e x µ σ

σ π

= where

µ= Mean of the normal random variable x

σ = Standard deviation 3.1416...

π =

2.71828...

e= ( )

P x<a is obtained from a table of normal probabilities

The student with knowledge of calculus should note that there is not a closed-form expression for ( ) b ( )

P a< <x b =

### ∫

a f x dx for the normal probability distribution . The value of this definite integral can be obtained to any desired degree of accuracy by numerical approximation

Chapter 4 Random Variables amd Probability Distribution

4.7 The Normal Distribution

P229

procedures . For this reason , it is tabulated for the user .

Sample Mean：(large n)

(1/ 2)[( ) / ]2

( ) 1

2

x x

x

f x e µ σ

σ π

=

Chapter 4 Random Variables amd Probability Distribution

Chapter Notes

p.273

(1) 區間 [ , ]a b 必須為有限。

(2) 被積分函數 f 在 [ , ]a b 必須為連續。或者，若不連續，也得在 [ , ]a b 中為有界。

(1) 對每一數 t≥ ，若 a t ( )

a f x dx

### ∫

a f x dx( ) =limt→∞

### ∫

at f x dx( )

(2) 對每一數 t b≤ ，若 b ( )

t f x dx

### ∫

−∞b f x dx( ) =tlim→−∞

tb f x dx( )

(3) 若 ( )

c f x dx

### ∫

−∞c f x dx( ) 皆為收斂，則稱廣義積分 f x dx( )

−∞ 為收斂，定義為

( ) f x dx

−∞ =

−∞c f x dx( ) +

### ∫

c f x dx( ) 。若此式等號右邊任一積分發散，則稱 f x dx( )

### ∫

−∞

H11、M19、M20、N32、N44、Q13、Q15、Q20、Q22，以上十四個學系採計數乙，建 議未來採計數甲；E09、H11、N16、N32，以上四個學系建議未來應規定學生於大一上 必修微積分。