04 บทที่2 การประมาณค่า 11 11 2556¹จงของต วอย...
TRANSCRIPT
-
2
208272 2 ..
2-1
1 N () () 1 2 nX ,X ,...,X n
ix i, i = 1, 2, 3, , n
n
ii 1 1 2 n
XX X XX
n n= + + += =
X (Estimator) (Estimate)
n
xxxn
xx n21
n
1ii +++==
=
2.1 (Sampling)
N 2
n n
XX
n
1ii
==
(Sampling with Replacement)
1 2 3 n-1 n N N N N N
nN
2
-
2
208272 2 ..
2-2
(Sampling without Replacement)
1 2 3 n-1 n N N-1 N-2 N-(n-2) N-(n-1)
nN C)!nN(!n!N
!n)1nN()2nN()2N()1N(N
=
=++
( k k !) 2.1 N = 4 1, 2, 3, 4
N
ii 1
X1 2 3 4 10 2.5
N 4 4= + + + = = = =
N
2 2i
i 1
1 (X )N =
=
25.145)25.225.025.025.2(
41
))5.1()5.0()5.0()5.1((41
))5.24()5.23()5.22()5.21((41
2222
2222
==+++=
+++=
+++=
n = 2 16Nn =
x )xX(P =
1 ( 1 , 1 ) 161
1.5 ( 1 , 2 ) ( 2 , 1 ) 162
2 ( 1 , 3 ) ( 2 , 2 ) ( 3 , 1 ) 163
2.5 ( 1 , 4 ) ( 2 , 3 ) ( 3 , 2 ) ( 4 , 1 ) 164
3 ( 2 , 4 ) ( 3 , 3 ) ( 4 , 2 ) 163
3.5 ( 3 , 4 ) ( 4 , 3 ) 162
4 ( 4 , 4) 161
1
-
2
208272 2 ..
2-3
X k
j jXj 1
1 2 1x P(X x ) 1 1.5 4 2.516 16 16= = = = + + + =
( )X1 1 1.5 1.5 2 4 2.5
16 = + + + + + =
X
2 2 2 2 2 2 2X
1 2 1 5(X ) ( (X)) 1 1.5 4 2.5 0.62516 16 16 8
= = + + + = =
( ) ( ) ( )k 16
2 2 22 2jX X
j 1
1 1 5(x ) 1 2.5 1.5 2.5 4 2.516 16 8
=
=
= = + + + =
X = 2
2X n
=
n = 2 6CC 242N ==
x )xX(P =
1.5 ( 1 , 2 ) 61
2 ( 1 , 3 ) 61
2.5 ( 1 , 4 ) ( 2 , 3 ) 62
3 ( 2 , 4 ) 61
3.5 ( 3 , 4 ) 61
1
X k
j jXj 1
1 1 1x P(X x ) 1.5 2 3.5 2.56 6 6=
= = = + + + =
( )X1 1.5 2 2.5 2.5 3 3.5 2.56
= + + + + + =
X
2 2 2 2 2 2 2X
1 1 1 40(X ) ( (X)) 1.5 2 3.5 2.5 6.256 6 6 6
20 25 80 75 5 0.416673 4 12 12
= = + + + =
= = = =
( ) ( ) ( )k 6
2 2 22 2jX X
j 1
1 1(x ) 1.5 2.5 2 2.5 3.5 2.56 6
=
=
= = + + +
[ ]125
25
61125.00025.01
61
==+++++=
X = 2
2X
N nN 1 n
=
#
-
2
208272 2 ..
2-4
2x
1NnN
1 2 2
2X
N n , nN 1 n n
=
2.2 (Point Estimation)
N
ii 1
X N=
= n
ii 1
X n X=
= =
n
ii 1
x n x=
= =
1x 1
2x 2
3x 3
2.3 (Interval Estimation)
L U < < L U (Confidence Limits) L U P( ) 1 < < = 0 1 1 (Confidence Coefficient) (1 )100% (Confidence Level) 1 L U ( , ) (Confidence Interval : CI)
1x 3x
-
2
208272 2 ..
2-5
1) 2) 3) ( )1 100% 4) L U ( , )
2.4
)X( 3 1) 2( ) 2)
(n 30)< 3) ( n 30 ) 2.4.1 2( )
X
XZ ~ N(0,1)= X
XZ ~ N(0,1)=
X
2 n
x = 2
2X n
=
22X
N nN 1 n
=
2XX ~ N ( , )
2
X ~ N ( , )n
XZ ~ N(0,1)n
=
( )1 100% z 1
/ 2 1
2
z1
2
z
/ 2
X
XZ ~ N(0,1)=
-
2
208272 2 ..
2-6
12 2
P z Z z 1
< < =
12 2
XP z z 1n
< < =
12 2
P z X z 1n n
< < =
12 2
P X z X z 1n n
+ < < + =
12 2
P X z X Z 1n n
< < =
1 12 2 2 2
P X z X z 1 ( z z )n n
+ < < + = =
2.2 0.2 15 ( 10,000 )
1.3 1.4 1.2 1.5 1.5 1.1 1.4 1.6
1.3 1.5 1.5 0.9 1.1 1.0 1.8 90%
( )1 100%
12 2
X z X zn n
+ < < + 1 1
2 2
X z , X zn n
+
1
2
X zn
1
2 2
x z x zn n
+ < < +
1 1
2 2
x z , x zn n
+
12
x zn
-
2
208272 2 ..
2-7
0.2, n 15, x 1.34 = = =
1 0.90 = 0.052= 645.1z
21
=
1
2
x zn
( ) ( )0.21.34 1.64515
=
( )1.255, 1.425=
1.255 10,000 1.425 10,000 90% #
2.3 1 10 16 69 1 95%
1
10, n 16, x 69 = = =
1 0.95 = 0.0252= 96.1z
21
=
1
2
x zn
( ) ( )16
1096.169 =
( )9.73,1.64=
1 64.1 73.9 95% # 2.4 16 95% 18.75 22.25
X : 2N( , )
-
2
208272 2 ..
2-8
XTS n
= n 1 =
X : 2
N( , )n
95% 18.75 22.25 18.75 22.25< <
1 1
2 2
x z x zn n
< < +
1 0.95 = 0.0252=
12
z 1.96
=
1 1
2 2
18.75 x z x z 22.25n n
= < < + =
18.75 x 1.96 x 1.96 22.2516 16
= < < + =
x 0.49 18.75 = (1)
x 0.49 22.25+ = (2)
(1) + (2) 2x 41= x 20.5=
(2) - (1) 0.98 3.5 = 3.5714 =
20.5 3.5714 = # 2.4.2 )30n( <
X n )30n( < )X(
(Degree of freedom : df or ) 4 x 3 1 2 3x , x , x 4 1 2 3(4 x) (x x x ) + + x 3 () 1 x n 1
-
2
208272 2 ..
2-9
2. 1
2.5 T
1) ( )10P T 1.372 p= < =
2) ( )15P T t 0.01= < =
3) ( )12P T t 0.05= > =
4 (Cumulative Students t Distribution) -13
1) ( )10P T 1.372 0.9= < = p 0.9=
2) ( )9P T 2.821 0.99= < = ( )9P T 2.821 0.01= < =
0.01,9t 2.821 =
0.01 0.01
0.01,9t 2.821= 0.99,9t 2.821=
3) ( )12P T 1.782 0.95= < =
( )12P T 1.782 1 0.95 0.05= > = =
0.05,12t 1.782 =
#
-
2
208272 2 ..
2-10
( )1 100%
/ 2 / 2
,2
t 1 ,
2
t
1
, 1 ,2 2
P t T t 1
< < =
, 1 ,2 2
XP t t 1S n
< < =
, 1 ,2 2
S SP t X t 1n n
< < =
, 1 ,2 2
S SP X t X t 1n n
+ < < + =
1 , ,2 2
S SP X t X t 1n n
< < =
, 1 , , 1 ,2 2 2 2
S SP X t X t 1 ( t t )n n
+ < < + = =
( )1 100%
, 1 ,2 2
S SX t X tn n
+ < < + 1 , 1 ,
2 2
S SX t , X tn n
+
1 ,
2
SX tn
n 1 =
, 1 ,
2 2
S Sx t x tn n
+ < < +
1 , 1 ,
2 2
S Sx t , x tn n
+
1 ,2
sx tn
n 1 =
-
2
208272 2 ..
2-11
XZn
=
XZS n
=
2.6 7
1 2 3 4 5 6 7 () 172 155 158 149 174 153 170
90%
s 10.18, n 7, x 161.57= = =
1 0.90 = 0.052= 0.05,6t 1.943=
1 ,
2
sx tn
( ) ( )10.18161.57 1.9437
=
( )154.094 , 169.046=
154.094 169.046 90% #
2.4.3 )30n(
(Central Limit Theorem) 2 n (n )
X = 2
2X n
=
n (n )
-
2
208272 2 ..
2-12
( 2 ) ( )1 100%
12 2
X z X zn n
+ < < + 1 1
2 2
X z , X zn n
+
1
2
X zn
1
2 2
x z x zn n
+ < < +
1 1
2 2
x z , x zn n
+
12
x zn
( 2 ) ( )1 100%
12 2
S SX z X zn n
+ < < + 1 1
2 2
S SX z , X zn n
+
1
2
SX zn
1
2 2
s sx z x zn n
+ < < +
1 1
2 2
s sx z , x zn n
+
12
sx zn
2.7 100 30 5 90%
s 5, n 100, x 30= = =
1 0.90 = 0.052= 0.95z 1.645=
-
2
208272 2 ..
2-13
1
2
sx zn
( ) 530 1.645100
=
( )29.1775 , 30.8225=
29.1775 30.8225 95% #
2.5 2
2 1 1 21 2 2 22 1 2 1 2X X =
1 2 1n 2n 1X 1 2X 2 21 XX 2
1 2 1 2X X = 2
1 2
2 22 1 2X X
1 2n n
= + 2
1) 2 2 21 2( , ) 2) 2 )30n,30n( 21
-
2
208272 2 ..
2-14
1 2 1 22 21 2
1 2
(X X ) ( )Z ~ N(0,1)
n n
=
+
1 2 ( )1 100%
2 2 2 21 2 1 2
1 2 1 2 1 2 11 2 1 22 2
(X X ) z (X X ) zn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 11 2 1 22 2
((X X ) z , (X X ) z )n n n n
+ + +
2 21 2
1 2 11 22
(X X ) zn n
+
2 2 2 21 2 1 2
1 2 1 2 11 2 1 22 2
(x x ) z (x x ) zn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 11 2 1 22 2
((x x ) z , (x x ) z )n n n n
+ + +
2 21 2
1 2 11 22
(x x ) zn n
+
( )1 100%
1 2 1 22 2 1
2 21 2
1 2
(X X ) ( )P(z z ) 1
n n
< < =
+
2 2 2 21 2 1 2
1 2 1 2 11 2 1 22 2
P(z (X X ) ( ) z ) 1n n n n
+ < < + =
2 2 2 21 2 1 2
1 2 1 2 1 2 11 2 1 22 2
P( (X X ) z ( ) (X X ) z ) 1n n n n
+ + < < + + =
2 2 2 21 2 1 2
1 2 1 2 1 211 2 1 22 2
P((X X ) z ( ) (X X ) z ) 1n n n n
+ < < + =
2 2 2 21 2 1 2
1 2 1 2 1 2 11 2 1 22 2
P((X X ) z ( ) (X X ) z ) 1n n n n
+ + < < + + =
-
2
208272 2 ..
2-15
( ) ( )1 2 1 22p
1 2
X XT
1 1Sn n
=
+
1 2n n 2 = +
( ) ( )2 2
1 1 2 22p
1 2
n 1 S n 1 SS
n n 2 +
=+
2.8 2 9 16 15 8 35 25 95%
1 2, 1 2 1 1 1n 15, x 35 , 9= = =
2 2 2n 8, x 25, 16= = =
1 0.95 = 0.0252= 0.975z 1.96=
95% 1 2 2 21 2
1 2 11 22
(x x ) zn n
+
2 29 16(35 25) (1.96)
15 8= +
10 11.9865= ( 1.9865 , 21.9865)=
-1.9865 21.9865 95% # 2.5.2 2 )30n,30n( 21
-
2
208272 2 ..
2-16
1 2 ( )1 100%
2 21 2 p 1 2 1 2 p, 1 ,
1 2 1 22 2
1 1 1 1(X X ) t S (X X ) t Sn n n n
+ + < < + +
2 21 2 p 1 2 p1 , 1 ,1 2 1 22 2
1 1 1 1((X X ) t S , (X X ) t S )n n n n
+ + +
21 2 p1 ,1 22
1 1(X X ) t Sn n
+
2 21 2 p 1 2 1 2 p, 1 ,
1 2 1 22 2
1 1 1 1(x x ) t s (x x ) t sn n n n
+ + < < + +
2 21 2 p 1 2 p1 , 1 ,1 2 1 22 2
1 1 1 1((x x ) t s , (x x ) t s )n n n n
+ + +
21 2 p1 ,1 22
1 1(x x ) t sn n
+
2.9
() () ()
16 8,500 1,000
9 12,500 1,200
95%
A B,
1 1 1n 16, x 8,500, s 1,000= = = 2 2 2n 9, x 12,500, s 1,200= = =
2 2 2 22 1 1 2 2p
1 2
(n 1)s (n 1)s 16(1,000) 9(1,2 1,259,130.4300)sn n 2 16 9 2
48 + += = =+ +
1 0.95 =
0.0252= 0.975,231 ,
2
t t 2.069
= =
-
2
208272 2 ..
2-17
( ) ( )1 2 1 22 21 2
1 2
X XT
S Sn n
=
+
( ) ( )
22 21 2
1 22 22 2
1 1 2 2
1 2
S Sn n
S n S nn 1 n 1
+
=
+
1 2 ( )1 100%
2 2 2 21 2 1 2
1 2 1 2 1 2, 1 ,1 2 1 22 2
S S S S(X X ) t (X X ) tn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 , 1 ,1 2 1 22 2
S S S S((X X ) t , (X X ) t )n n n n
+ + +
2 21 2
1 2 1 ,1 22
S S(X X ) tn n
+
2 2 2 21 2 1 2
1 2 1 2 1 2, 1 ,1 2 1 22 2
s s s s(x x ) t (x x ) tn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 , 1 ,1 2 1 22 2
s s s s((x x ) t , (x x ) t )n n n n
+ + +
2 21 2
1 2 1 ,1 22
s s(x x ) tn n
+
95% 1 2 2
1 2 p1 ,1 22
1 1(x x ) t sn n
+
1, 2 1 1(8 59,1,500 12, 30.4348500) (2.069)16 9 = +
967.35224,000= ( 4,967.3522 , 3,032.6478)=
-4,967.3522 -3,032.6478 95% #
2.5.2.2 2 21 2
-
2
208272 2 ..
2-18
1 2 1 22 21 2
1 2
(X X ) ( )Z ~ N(0,1)
n n
=
+
2.10 2 1 16 2 12 62 54 4 7 95%
A B, 1 2 1 1 1n 16 , x 54, s 4= = =
2 2 2n 12, x 62, s 7= = =
1 0.95 = 0.0252=
( ) ( ) ( ) ( )
2 22 2 2 21 2
1 22 2 2 22 2 2 2
1 1 2 2
1 2
s s 4 7n n 16 12
17s n s n 4 16 7 12n 1 n 1 16
16.3292
1 12
9
1
+ +
= = =
+ +
0.975,171 ,2
t t 2.110
= =
95% A B 2 2A B
A B 1 ,A B2
s s(x x ) tn n
+
2 24 7(62 54) (2.110)
16 12= +
8. 48 870= ( 0.8704 , 16.8704)=
-0.8704 16.8704 95% # 2.5.3 1 2(n 30 , n 30)
1n 2n 1 2 21 XX
2.5.3.1 2 21 2,
-
2
208272 2 ..
2-19
1 2 ( )1 100%
2 2 2 21 2 1 2
1 2 1 2 1 2 11 2 1 22 2
(X X ) z (X X ) zn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 11 2 1 22 2
((X X ) z , (X X ) z )n n n n
+ + +
2 21 2
1 2 11 22
(X X ) zn n
+
2 2 2 21 2 1 2
1 2 1 2 11 2 1 22 2
(x x ) z (x x ) zn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 11 2 1 22 2
((x x ) z , (x x ) z )n n n n
+ + +
2 21 2
1 2 11 22
(x x ) zn n
+
1 2 1 22 21 2
1 2
(X X ) ( )Z ~ N(0,1)S Sn n
=
+
1 2 ( )1 100%
2 2 2 21 2 1 2
1 2 1 2 1 2 11 2 1 22 2
S S S S(X X ) z (X X ) zn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 11 2 1 22 2
S S S S((X X ) z , (X X ) z )n n n n
+ + +
2 21 2
1 2 11 22
S S(X X ) zn n
+
2.5.3.2 2 21 2, 21S 22S
-
2
208272 2 ..
2-20
2 2 2 21 2 1 2
1 2 1 2 1 2 11 2 1 22 2
s s s s(x x ) z (x x ) zn n n n
+ + < < + +
2 2 2 21 2 1 2
1 2 1 21 11 2 1 22 2
s s s s(x x ) z , (x x ) zn n n n
+ + +
2 21 2
1 2 11 22
s s(x x ) zn n
+
2.11
2 30 40 4.5 0.6 4.2 0.4 2 95%
1 2, 1 2
1 1 1n 30, x 4.5, s 0.6= = =
2 2 2n 40, x 4.2, s 0.4= = =
1 0.95 = 025.02=
96.1z 975.0 =
95% 1 2 2 21 2
1 2 11 22
s s(x x ) zn n
+
2 20.6 0.4(4.5 4.2) (1.96)
30 40= +
0.3 0.247923=
(0.052077 , 0.547923)=
2 0.052077 0.547923 95% #
-
2
208272 2 ..
2-21
D
D
DTS n
= n 1 =
2.6 2 2
)Y,X( XYD = 1 Y 2
X D Y X = n )Y,X( ii iD iii XYD =
n
DD
n
1ii
== n
1n
)DD(S
n
1i
2i
D
== n
D
N
ii 1
Y X DD
D
N= = = =
2
2 DD n
= ,
N2
i Di 1
D
(D )
N=
=
n )D(
-
2
208272 2 ..
2-22
D ( )1 100%
D DD, 1 ,
2 2
S SD t D tn n
+ < < + D D1 , 1 ,
2 2
S S(D t , D t )n n
+
D1 ,
2
SD tn
d dD, 1 ,2 2
s sd t d tn n
+ < < + d d1 , 1 ,
2 2
s sd t , d tn n
+
d1 ,
2
sd tn
n
dd
n
1ii
==
n2
ii 1
d
(d d)s
n 1=
=
2.12 10
)y( i )x( i
1 18.5 15 2 28.5 25
3 17 18.5
4 13.5 11
5 14.5 10
6 16 12
7 10 11.5
8 19.5 15.5
9 16 11
10 20 25.5
90%
D
-
2
208272 2 ..
2-23
iy ix iii xyd = 1 18.5 15 3.5 2 28.5 25 3.5
3 17 18.5 -1.5
4 13.5 11 2.5
5 14.5 10 4.5
6 16 12 4
7 10 11.5 -1.5
8 19.5 15.5 4
9 16 11 5
10 20 25.5 -5.5
n2
ii 1
d
(d d)n 10, d 1.85, s 3.4725
n 1=
= = = =
1 0.90 = 0.052= 0.95,91 ,n 1
2
t t 1.833
= =
90% D d1 ,n 12
sd tn
(3.4725)1.85 (1.833)10
=
( 0.1628 , 3.8628)=
-0.1628 3.8628 90% #
2.7 1
2 P
N
XP
N
1ii
== N
iX
1 i
0 i =iX
-
2
208272 2 ..
2-24
P E(P) P PZ V(P) P(1 P)
n
= =
P ( ) %1001
1
2 2
P(1 P) P(1 P) P z P P zn n
+ < < +
1 1
2 2
P(1 P) P(1 P) (P z , P z )n n
+ 1
2
P(1 P)P zn
1
2 2
p(1 p) p(1 p) p z P p zn n
+ < < +
1 1
2 2
p(1 p) p(1 p) (p z , p z )n n
+ 12
p(1 p)p zn
n
ii 1
xxp
n n== =
x
n
nX
n
XP
n
1ii
=== X
n
iPi 1
nPE(P) E X n Pn=
= = = =
n n
2 i i2 2P
i 1 i 1
1 nP(1 P) P(1 P)V(P) V( X n) V( X )n n n= =
= = = = =
n 2P P(1 P)
n
= 2P
-
2
208272 2 ..
2-25
2.13 1,000 850 95% P
850n 1000, p 0.851000
= = =
1 0.95 = 0.0252= 0.975z 1.96=
95% P 1
2
p(1 p)p zn
( ) (0.85) (0.15)0.85 1.961000
=
0.85 0.022132= (0.827868,0.872132)=
0.827868 0.872132 95% #
2.8 2
2
1
N
1ii1
1 N
XP
1
== 1
2
N
1ii2
2 N
XP
2
== 2
i1X i2X
1 i 1
0 i 1
1 i 2
0 i 2
21 P,P
=i1X
=i2X
-
2
208272 2 ..
2-26
2
22
1
11
2121
2
22
1
11
2121
n)P1(P
n)P1(P
)PP()PP(
n)P1(P
n)P1(P
)PP()PP(Z
+
+
=
21 PP ( )1 100%
1 1 2 2 1 1 2 21 2 1 2 1 2 1
1 2 1 22 2
P (1 P ) P (1 P ) P (1 P ) P (1 P ) (P P ) z P P (P P ) zn n n n
+ + < < + +
1 1 2 2 1 1 2 21 2 1 21 11 2 1 22 2
P (1 P ) P (1 P ) P (1 P ) P (1 P ) ((P P ) z , (P P ) z )n n n n
+ + +
1 1 2 21 2 11 22
P (1 P ) P (1 P ) (P P ) zn n
+
1
n
1ii1
1 n
XP
1
== 1
2
n
1ii2
2 n
XP
2
== 2
( 21 PP ) ( 21 PP )
1 2
1 2 1 2P P E(P P ) P P
= =
1 2
2 1 1 2 2 1 2P P
1 2
P (1 P ) P (1 P ) V(P P )n n
= = +
21 n,n
2
22
1
11
2121
n)P1(P
n)P1(P
)PP()PP(Z
+
=
21 n,n ( 21 PP )
( 21 PP ) 1 2
2 1 1 2 2 P P
1 2
P (1 P ) P (1 P )n n
= + 1 2
2 P P
-
2
208272 2 ..
2-27
1 1 2 2 1 1 2 21 2 1 2 1 2 1
1 2 1 22 2
p (1 p ) p (1 p ) p (1 p ) p (1 p ) (p p ) z P P (p p ) zn n n n
+ + < < + +
1 1 2 2 1 1 2 21 2 1 21 11 2 1 22 2
p (1 p ) p (1 p ) p (1 p ) p (1 p ) ((p p ) z , (p p ) z )n n n n
+ + +
1 1 2 21 2 11 22
p (1 p ) p (1 p ) (p p ) zn n
+
1n
1ii 1 1
11 1
xxp
n n== =
2n
2ii 1 2
22 2
xxp
n n== =
21 x,x
1 2
2.14 95% 21 P,P
1 11,830n 2,000, p 0.9152,000
= = =
2 22,150n 2,500, p 0.862,500
= = =
1 0.95 = 0.0252= 0.975z 1.96=
95% 21 PP
1 1 2 21 2 11 22
p (1 p ) p (1 p ) p p zn n
+
( ) (0.915) (0.085) (0.86) (0.14)(0.915 0.86) 1.962,000 2,500
= +
0.055 0.018=
2,000 1,830 2,500 2,150
-
2
208272 2 ..
2-28
2 2n
2 i2 2
i 1
(X X) (n 1)S=
= =
n 1 =
(0.037, 0.073)=
0.037 0.073 95% # 2.9
2 2 n 2S 2 2E(S ) =
2
2. 2
0
2.15 2
1) ( )214P 29.1 p < = 2) ( )210P c 0.05 < = 3) ( )212P c 0.001 > =
5 (Cumulative Chi-square Distribution) -14-17
1) ( )214P 29.1 0.99 < = p 0.99=
-
2
208272 2 ..
2-29
2 ( )1 100%
2 22
2 2
1 ,n 1 ,n 12 2
(n 1)S (n 1)S