Cong thuc XSTK full_Hoso Lee.pdf

Hc thng minh ng l mt sch [email protected] https://www.facebook.com/pages/Notbookworms/561803483864811

L Hng Sn Anh 17 KTN K49 FTU 0166.8589.667 https://www.facebook.com/hosolee.superdevil [email protected]

Phn I:

A.B = V xung khc (i chi vs i hc)

.A B UA B V

A B

B A

i (True or False)

P(A) khng nh hng n P(B) -> c lp

lim ( ) ( )nn f A P A

XS c iu kin

P (A|B) = ( )( )

P ABP B

( P(B) >0 )

P (AB) = P(A|B) . P(B)

.A B A B .A B A B P(A+B+C) = P(A) + P(B) + P(C) P(AB) P(BC) P(AC) + P(ABC)

XS y :

P(A) = 1

( | ). ( )n

i ii

P A B P B

Bayes:

P (Bi|A) = ( | ). ( )

( )i iP A B P B

P A

Phn II.

P (X=XK) = PK P (a X < b) = 2i ix p ; a XK < b Hm mxs: f(x) = F(x) f(x) 0 x v ( )f x dx

= 1 P (a X < b) = ( )b

a

f x dx = F(b) F(a) Hm pbxs: F(x) [0;1] x R F(x) = ( )x f x dx

F(x) l hm khng gim

E (X) = i ix p = ( )xf x dx

E (g(x)) = . ( )ip g x = ( ). ( )g x f x dx

E (X2) = 2i ix p = 2 ( )x f x dx

V (X) = E (X2) E2 (X)

Mod X = x0 f(x) t GTLN ti x = x0 Med X = xi F (xi) 0.5 < F (xi+1) Med X = x0 F (x0) = 0.5 Nh thc X ~ B (n,p): PK = (1 )n k n kkC p p

E (X) = np V (X) = npq np q Mod (X) np +q Siu bi: P (X=m) =

.m n mM N MnN

C CC

E (X) = .n MN

Var (X) = . . .1

M N M N nnN N N

= npq. 1

N nN

Poisson PK = .!

kek

( > 0)

E (X) = Var (X) = Mod X = -1 v nu nguyn Mod X = [] nu khng nguyn 1



Ly tha X ~ E () nu f(x) = 0 x < 0 F (x) = 0 x < 0 = . e-x x 0 = 1 e-x x 0 E (X) = 1/ Var (X) = 1/2 1 Mod X Pp u trn [a,b]

f (x) = 0 [ , ]1 [ , ]

x a b

x a bb a

F (x) =

0

1

x ax a a x bb a

x b

E (X) = 2

a b Var (X) =

2( )12

b a

Chun X ~ N (;2) 0 () = 0.5 0 (-u) = - 0 (u) u1- = -u P (a x < b) = 0

b

- 0 a

P (|X-| < ) = 95.44% 299.74% 3

(kx) ~ N (k; k22) (x+a) ~ N (+a; 2) 2 bin c lp X1 + X2 ~ N (1 + 2; 2 21 2 ) BNN 2 chiu P (x1 X x2 , y1 Y y2) = F(x1,y1) + F(x2,y2) - F(x1,y2) - F(x2,y1) f (x,y) = 2 ( , )

.F x yx y

F (x,y) = ( , ) .yx

f x y dx dy

f1 (x) = ( , )f x y dy f2 (y) = ( , )f x y dx

X; Y c lp ri rc: Pij = P(xi) . P(yj) Lin tc: f(x,y) = f1(x). f2(y) F(x,y) = F1(x). F2(x) Cov (X;Y) = E(XY) E(X). E(Y) Ri rc: E (XY) = xi yj pij Lin tc: E (XY) = . ( , ) .xy f x y dx dy

H s tng quan ( ; )

.xy x y

Cov X Y

|xy| 1 c lp: xy = 0

Ch : Khi nh thc c n 20 v p 0.1 AD cng thc Poisson vi = np

Khi nh thc c n > 5 v 1 1 0.3

1p p

p p n

AD cng thc pp Chun vi = np v 2 = npq

P (X = x) = 1 x npnpq npq

P (a X b) = 0 0b np a npnpq npq 2

C LNG : X ~ N(, 2) Khi bit 2 Khi cha bit 2

L trung bnh tng th N(0,1)~)(

XU 1)-T(n~)(S

nXT

Khong tin cy i xng22

Un

XUn

X )1(2

)1(

2

nn tn

SXtn

SX

di tin cy2

U

n ; I = 2 )1(

2

ntn

S

; I = 2

Khong tin cy ti a

Un

X )1( ntn

SX

Khong tin cy ti thiu

U

nX )1( nt

nSX

Kch thc mu o

oUn

2

22/ )

.('

o

Un

)1(

2

ntn

S

o2

)1(2/ ).('

o

ntSn

di tin cy i xng I Io

22/ ).2

('o

Un

2)1(

2/ ).2('o

n

ItSn

Khi bit Khi cha bit

L phng sai tng th 2 )(22

2*2 ~ nnS

))1((22

22 ~)1( nSn

Vi tin cy 1- cho trcKhong tin cy 2 pha

)(2

21

2*2

)(2

2

2*

nn

nSnS

)1(2

21

22

)1(2

2

2 )1()1(

nn

SnSn

Khong tin cy ti a )(21

2*2

n

nS

)1(21

22 )1(

n

Sn

Khong tin cy ti thiu )(2

2*2

n

nS

)1(2

22 )1(

n

Sn

Khong tin cy ti a:

U)1(

nff

fp

Khong tin cy ti thiu:

U)1(

nff

fp

Xc nh kch thc mu

L t l tng th

N(0,1)~)1(

)(ffnpfU

Khong tin cy i xng

22

U)1(

U)1(

nff

fpn

fff

2

U)1(

nff

di khong tin cy x:I = 2

o2

0

2/.U)1('

ff

n

I Io 2

0

2/.U)1(2'

ff

n

Sai s ca c lng

Sai s ca c lng

[email protected]://www.facebook.com/pages/Notbookworms/561803483864811

c thng minh H ng l mt sch

L H ng Sn Anh 17 K49 FTU https://www.facebook.com/hosolee.superdevil hosolee1@

KTN gmail.com

Phn III:

3



K hiu trong kq ca my tnh CASIO: S* = xn (my 570 ES) = x (570 ES plus) S = xn-1 = s

T BNN gc X c E (X) = m v V (X) = 2, mu nn kch thc n: E ( X ) = m ; V ( X ) = Se2( X ) = 2n

* Trung v: + Ri rc: n chn Xd l 2 gi tr chnh gia n l: Xd = gi tr th 1

2n

+ Ghp lp: Xd = L + 0.5.

dX

n Shn

L: gii hn di lp cha trung v n: kch thc mu S: tng tn s lp ng trc lp cha trung v

dXn : tn s ca lp cha trung v

h: di lp cha trung v.

* Mt: X0 = gi tr tn s ln nht Nu ghp lp: X0 = L + h. 11 2

dd d

L: Gii hn di ca lp cha mt n: di lp cha mt d1: tn s lp cha mt tn s lp ng trc d2: tn s lp cha mt tn s

lp ng sau

* H s bin thin: CV = 100. SX

L 2 tham s ca BNN

Hiu 2 k vng nh 1 tham s: thay X -> 1 2X X v -> 1 2

* bit 2 21 2; : thay n

-> 2 21 2

1 2n n

* Cha bit 2 21 2;

coi 2 21 2 : thay (n-1) -> n1+n2-2 v Sn

-> SP. 1 2

1 1n n vi SP =

2 21 1 2 2

1 2

( 1). ( 1).2

n S n Sn n

coi 2 21 2 : thay Sn

-> 2 21 2

1 2

S Sn n

v (n-1) -> 1 22 22 1

( 1)( 1)( 1) (1 ) ( 1)

n nn C C n

vi

21

12 2

1 2

1 2

SnC

S Sn n

Hiu 2 xc sut nh 1 tham s: thay p -> p1 p2 Thay (1 )f f

n

-> Sf = 1 1 2 21 2

(1 ) (1 )f f f fn n

4



T s 2 phng sai Thng k F = 2 22 12 21 2

.SS

~ F (n2 1; n1 1)

KTC 2 pha: P ( 2 12

( 1; 1)12 12 2

. n nS fS

<

2122

< 2 12

( 1; 1)122 2

. n nS fS

) = 1 , ,1

1n mn mf f

KTC bn phi: P ( 2 1

2( 1; 1)1

122

. n nS fS

<

2122

) = 1 KTC bn tri: P ( 2122

< 2 12

( 1; 1)122

. n nS fS

) = 1 Phn IV: KIM NH

Cc quy tc cn nh

1. H0 lun lun c du =

2. C 2 cch nhn xt H0, H1, nhng phi ty thuc vo yu cu bi a ra kt lun.

Nu qs W bc b H0 Chp nhn H1

Nu qs W cha c c s bc b H0 Bc b H1

Cc dng bi thng dng

Dng 1. ni l nh nhau (khng thay i, khng khc nhau) H0: a = b H1: a b Nu qs W bc b H0 Kt lun: l khc nhau (thay i) Nu qs W cha c c s bc b H0 Kt lun: cha c c s bc b l khc nhau (thay i)

Dng 2. ni l c s khc nhau (thay i) H0: a = b H1: a b Nu qs W chp nhn H1 Kt lun: c s khc nhau (thay i) Nu qs W bc b H1 Kt lun: k c s khc nhau (thay i)

Dng 3. ni l a > b (a b (a b (a b (a

KIM NH X ~ N(, 2)Khi bit 2 Khi cha bit 2

Kim nh trung bnh ()Tiu chun K: )( n

XU

)(S

nXT

Ho: = oH1: o

W = (-,-U/2) (U/2, +) ),(),( )1( 2/)1(

2/ nn ttW

Ho: = oH1: > o

W = (U, +) ),( )1( ntW Ho: = oH1: < o

W = (-,-U) ),( )1( ntW

Kim nh phng sai (2) Kim nh t l tng th

Tiu chun K: 2

22 )1(

o

Sn

Tiu chun K: N(0,1)~)1(

)( 0oo ppnpfU

Ho:22o

H1:22o

),(),0( )1(22

)1(2

21

nnW

Ho: p = poH1: p po

),(),( 2/2/ UUW

Ho:22o

H1:22o

),( )1(2 nW

Ho: p = poH1: p > po

),(

UW

Ho:22o

H1:22o

),0( )1(21

nW

Ho: p = poH1: p < po

),(

UW

Kim nh 2 trung bnhX1 ~ N(1,

21 ) ; X2 ~ N(2,

22 )

Kim nh 2 t lX1 ~ N(1,

21 ) ; X2 ~ N(2,

22 )

TCK: N(0,1)~

2

22

1

21

21

nS

nS

XXU

TCK:

)11)(1(21

21

nnff

ffU

1

11 n

mf ;2

22 n

mf ;21

21

nnmmf

Ho: 1 = 2H1: 1 2

W = (-,-U/2) (U/2, +)Ho: p1 = p2H1: p1 p2

),(),( 2/2/ UUWHo: 1 = 2H1: 1 > 2

W = (U, +)Ho: p1 = p2H1: p1 > p2

),(

UWHo: 1 = 2H1: 1 < 2

W = (-,-U)Ho: p1 = p2H1: p1 < p2

),(

UW

Kim nh phng sai hai tng th X1 ~ N( 21 1, ) , X2 ~ N( 22 2, )

Kim nh tnh c lp ca 2 bin nh tnh . .

1 . .

: c l p

: ph thu coH X v Y

H X v Y u

Tiu chun kim nh:

2ij2

,( 1)i j i j

nn

n m

2( 1).( 1)( ; )k hW

Tiu chun kim nh: 2

122

SFS

2 21 22 2

1 1 2

:

:oH

H

1 2( 1; 1)( ; )n nW f

2 21 22 2

1 1 2

:

:oH

H

1 2( 1; 1)

1(0; )n nW f

Kim nh bin ngu nhin c phn phi chun

, ?

, ?

1

: i chu n

: i chu n

oH X c ph n ph

H X kh ng c ph n ph

Tiu chun kim nh:

2 23 4( 3)[ ]6 24a a

JB n

3 4

3 43 4

( ) ( )1 1;i i i ii i

x x n x x na a

n ns s

2(2)( ; )W

2 21 22 2

1 1 2

:

:oH

H

1 2 1 2( 1; 1) ( 1; 1)

12 2

(0; ) ( ; )n n n nW f f

[email protected]://www.facebook.com/pages/Notbookworms/561803483864811

c thng minh H ng l mt sch

L H ng Sn Anh 17 K49 FTU https://www.facebook.com/hosolee.superdevil hosolee1@

KTN gmail.com

6



Mnh s c gng ht sc c th gii p mi thc mc lin quan n mn hc XSTK ny cho cc bn. Cc bn lin lc vi mnh qua: - email [email protected] - facebook https://www.facebook.com/hosolee.superdevil

Ngoi ra, trong nm hc, c 2 thng, mnh s m lp t 20 25 ngi/ lp n thi mn ny. Nu bn no gp kh khn trong vic t hc, bn c th lin h vi mnh ng k lp hc. Chc cc bn t kt qu cao trong mn hc ny ^^

7

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