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Tm tt L thuyt xc sut
TS. L Xun Trng
Khoa Ton Thng k
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 1 / 32
1. Khng gian Xc sut
Mt khng gian xc sut l mt b ba (,F ,P), trong Khng gian mu : tp hp tt c cc kt qu c kh nng xy raca mt th nghim hay hin tng ngu nhin.
Khng gian cc bin c F : mt -i s cc tp con ca , tc lF 2 tha cc iu kin:
- F- A 2 F ) Ac 2 F- Nu Ai 2 F , 8i 2 N th [i=1Ai 2 F .
o Xc sut: mt hm P : F ! R tha mn cc tnh cht sau- P (A) 0, 8A 2 F- P () = 1- Nu A1,A2, ... l cc bin c ri nhau (Ai \ Aj = , 8 i 6= j) th
P ([i=1Ai ) = iP (Ai ) .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 2 / 32
1. Khng gian Xc sut
V d 1: Xt th nghim "tung mt xc sc cn i v ng cht gm 6mt"
Khng gian mu: = f1, 2, 3, 4, 5, 6g .C th xc nh nhng khng gian cc bin c khc nhau trn cngkhng gian mu
- F0 = f,g- F1 = 2
o xc sut
- Trn F0: hm P : F0 ! R cho bi P () = 0 v P () = 1.- Trn F1: hm P : F1 ! R cho bi
P (A) =i
6,
vi i l s phn t ca A 2 F1.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 3 / 32
2. Bin ngu nhin
Cho (,F ,P) l mt khng gian xc sut. Mt hm s : ! Rsao cho vi mi 2 R, ta c
f g := f 2 : () g 2 F ,
c gi l mt bin ngu nhin (hay mt hm F -o c) trn .
V d 2: Gi s A 2 F . Khi hm s IA () =
1, 2 A,0, /2 A, l
mt bin ngu nhin trn v nu 2 R, ta c
f 2 : IA () g =
8
3. Sigma i s sinh bi bin ngu nhin
Gi s : ! R l mt bin ngu nhin. Ta k hiu () l h ttc cc tp con ca c dng
1 (B) := f 2 : () 2 Bg ,trong B l mt tp Borel ca R (B 2 B(R)). Khi () l mtsigma i s trn v ta gi l sigma i s sinh bi bin ngu nhin.
V d 3: Xc nh sigma i s c sinh bi bin ngu nhin IA ()cho trong V d 2.Gii Xt B 2 B(R). Khi
I1A (B) =
8>>>>>:, nu 0, 1 /2 B ,A, nu 1 2 B v 0 /2 B ,Ac , nu 0 2 B v 1 /2 B ,, nu 0, 1 2 B .
Vy (IA) = f,,A,Acg .TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 5 / 32
4. Phn b xc sut ca bin ngu nhin
Xt khng gian xc sut (,F ,P) v : ! R l mt bin ngu nhin.Ta c th xc nh mt o xc sut P xc nh trn B(R) nh sau
P (B) = P1 (B)
.
Ta gi P l phn b xc sut ca bin ngu nhin .
Hm s F : R ! [0, 1] xc nh bi
F (x) = P ((, x ]) = P ( x) , x 2 R,
c gi l hm phn b xc sut tch ly ca .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 6 / 32
5. Hm mt ca bin ngu nhin
Cho l mt b.n.n xc nh trn khng gian xc sut (,F ,P).Nu tn ti mt dy s thc i mt khc nhau x1, x2, ... sao cho
P (B) = P ( 2 B) = xi2B
P ( = xi ) ,
vi mi B 2 B(R), th ta ni c phn b ri rc vi cc gi trx1, x2, ... v mt f (xi ) := P ( = xi ) ti xi .
Nu tn ti mt hm kh tch Lebesgue f : R ! R sao cho
P (B) = P ( 2 B) =Z
Bf (x) dx ,
vi mi B 2 B(R), th ta ni c phn b lin tc tuyt i v fc gi l hm mt ca .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 7 / 32
6. Cc c trng ca bin ngu nhin
Cho l mt b.n.n xc nh trn khng gian xc sut (,F ,P).K vng (Expectation):
E () =
8
6. Cc c trng ca bin ngu nhin
Lu :
Trong nh ngha k vng, ta hiu rng tng
xP ( = x) hocZ +
xf (x) dx
phi tn ti hu hn. Lc ta ni kh tch.
E (IA) = P (A)
Nu c moment bc k th s c mi moment bc b hn k .
Nu Ejjk
< th xc sut ui (tail probability) hi t v 0, tc
l
limn!+
P (jj > n)nk
= 0.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 9 / 32
6. Cc c trng ca bin ngu nhin
Mt s tnh cht
E (a) = a, 8a 2 R.E (a1 + b2) = aE (1) + bE (2)
Var (a) = 0, 8a 2 R.Var (a1) = a
2Var (1) , Var (1 + 2) = Var (1) + Var (2)
Nu l mt bin ngu nhin v g : R ! R l mt hm o c th
E (g ()) = g (x)P ( = x) , khi c phn b ri rc,
E (g ()) =R + g (x) f (x) dx , khi c phn b lin tc.
V d 4: Gi s l bin ngu nhin c hm mt xc nh bi
P ( = 0) = 0, 2; P ( = 1) = 0, 5; P ( = 2) = 0, 3.
Tnh E () v E2.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 10 / 32
7. Mt s phn b thng gp
Phn b Hm mt K vng Phng sai
B (n; p) P ( = k) = C kn pk (1 p)nk np np (1 p)
P() P ( = k) = e kk !
U (a; b) f (x) =
1ba , x 2 (a, b)0, x /2 (a, b)
a+b2
(ba)212
Exp () f (x) =
ex , x 0,0, x < 0.
1
12
N; 2
f (x) = 1
p
2e
(x)222 2
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 11 / 32
8. Phn b ng thi v phn b bin
Cho 1, 2 l hai b.n.n xc nh trn (,F ,P).Vi A,B 2 B(R), ta t
P12 (A B) = P (1 2 A, 2 2 B) .
Khi P12 l o xc sut trn B(R2) v gi l phn b xc sutng thi ca 1, 2.
Hm phn b xc sut tch ly ng thi ca hai bin ngu nhin 1v 2 c xc nh nh sau
F12 (x , y) = P (1 x , 2 y) .
Hm phn b xc sut tch ly bin
- theo 1 : F1 (x) = P12 ((, x ]R) = P (1 x , 2 +) .- theo 2 : F2 (y) = P12 (R (, y ]) = P (1 +, 2 y) .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 12 / 32
9. Hm mt ng thi - hm mt bin
Gi s 1 v 2 l hai bin ngu nhin c phn b ri rc v
1 2 fx1, x2, ...g , 2 2 fy1, y2, ...g .
Phn b xs ng thi ca 1, 2 cho bi
P12 (A B) = xi2A,yj2B
P (1 = xi , 2 = yj ) .
Ta gi f12 (x , y) = P (1 = x , 2 = y) l hm mt ng thi.
Hm mt bin theo 1 v 2 ln lt xc nh bi
f1 (xi ) = jP (1 = xi , 2 = yj ) ,
f2 (yj ) = iP (1 = xi , 2 = yj ) .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 13 / 32
9. Hm mt ng thi - hm mt bin
Nu tn ti mt hm kh tch Lebesgue f12 : R2 ! R sao cho
P12 (A B) =RR
AB f12 (x , y) dxdy ,
th ta ni 1 v 2 c phn phi xs ng thi lin tc tuyt i. Hmf12 c gi l hm mt xc sut ng thi ca 1, 2.
Hm mt xc sut bin theo 1 v 2
f1 (x) =Z +
f12 (x , y) dy ,
f2 (y) =Z +
f12 (x , y) dx .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 14 / 32
10. Hm ca cc bin ngu nhin
Cho 1 v 2 l hai b.n.n trn khng gian (,F ,P) c hm mt ngthi l f12 (x , y) v g : R
2 ! R l hm Borel. Khi
g (1, 2) l mt bin ngu nhin trn (,F ,P).K vng
- nu 1 v 2 ri rc
E (g (1, 2)) = x
yg (x , y) f12 (x , y)
- nu 1 v 2 lin tc
E (g (1, 2)) =Z +
Z +
g (x , y) f12 (x , y) dxdy
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 15 / 32
10. Hm ca cc bin ngu nhin
Cho 1 v 2 l hai b.n.n trn khng gian (,F ,P) c hm mt ngthi l f12 (x , y). Gi s
1 = g1 (X ,Y ) , 2 = g2 (X ,Y ) ,
trong X ,Y l hai b.n.n v nh thc Jacobi
J =
g1X g1Yg2X
g2Y
6= 0.Khi hm mt ng thi fXY (x , y) ca X ,Y tha mn
fXY (x , y) = f12 (g1 (x , y) , g2 (x , y)) jJ j .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 16 / 32
11. Mt s v d
V d 5: Cho hai bin ngu nhin ri rc X ,Y vi hm mt xc sutng thi fXY xc nh bi
fXY (1, 1) = P (X = 1,Y = 1) = 0, 5
fXY (1, 2) = P (X = 1,Y = 2) = 0, 1
fXY (2, 1) = P (X = 2,Y = 1) = 0, 1
fXY (2, 2) = P (X = 2,Y = 2) = 0, 3
Tnh xc sut P (X = 1) v P (Y = 1) .
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 17 / 32
11. Mt s v d
V d 6: Cho hai bin ngu nhin X ,Y vi hm mt ng thi
fXY =
6xy (2 x y) , (x , y) 2 (0, 1) (0, 1) ,0, (x , y) /2 (0, 1) (0, 1) .
Tm fX (x) v fY (y) .
V d 7: Cho hai bin ngu nhin X ,Y c hm mt ng thi
fXY =
x + y , (x , y) 2 (0, 1) (0, 1) ,0, (x , y) /2 (0, 1) (0, 1) .
Tnh E (X ) ,E (Y ) ,Var (X ) ,Var (Y ) v cov (X ,Y ) .
(cov (X ,Y ) = E ((X E (X )) (Y E (Y )))= E (XY ) E (X ) .E (Y ))
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 18 / 32
11. Mt s v d
V d 8: Cho hai bin ngu nhin X ,Y vi hm mt ng thi
fXY (x , y) =
x + y , 0 x y 1,0, otherwise.
t Z = X + Y . Xc nh hm mt v hm phn phi tch ly ca Z .
V d 9: Gi s X ,Y l hai bin ngu nhin c hm mt ng thi l
fXY (x , y) =
ex1 , 0 x2 x1 < +,0, otherwise.
t Z1 = X + Y v Z2 = X Y . Tm hm mt ng thi ca Z1,Z2v hm phn b tch ly bin ca Z2.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 19 / 32
12. Xc sut c iu kin. Tnh c lp
Cho A,B 2 F sao cho P (B) 6= 0. Xc sut c iu kin ca A khicho trc B xc nh bi
P (AjB) = P (A\ B)P (B)
.
Hai bin c A,B 2 F c gi l c lp nu
P (A\ B) = P (A) .P (B)
Hai bin ngu nhin v c gi l c lp nu vi mi A,B 2B(R) ta c
f 2 Ag v f 2 Bgl hai bin c c lp.
Ta ni hai sigma i s F1 v F2 l c lp nu 8A 2 F1, 8B 2 F2th A,B c lp.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 20 / 32
12. Xc sut c iu kin. Tnh c lp
Theorem
Nu v l hai bin ngu nhin kh tch th chng khng tng quan,tc l
E (.) = E ()E () hay suy ra cov (, ) = 0.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 21 / 32
13. Hm sinh monment
Hm sinh moment ca bin ngu nhin c xc nh bi
M (t) = Eet,
nu k vng tn ti. Nh vy, nu f l hm mt ca th
M (t) = x etx f (x) , khi ri rc,
M (t) =R + e
tx f (x) dx , khi lin tc.
Ch :
- Hm sinh moment cho php xc nh cc moment Mk () = E
k.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 22 / 32
13. Hm sinh monment
Theorem
Nu M (t) hu hn trn mt khong m cha 0 th hm t 7! M (t) khvi mi cp v
M(k) (t) = E
et .k
.
H qu l
M(k) (0) = E
k= Mk () .
Chng minhddtM (t) =
ddt
R + e
tx f (x) dx =R + xe
tx f (x) dx = Eet
,
d2
dt2M (t) =
ddt
R + xe
tx f (x) dx =R + x
2etx f (x) dx = E2et
,
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 23 / 32
13. Hm sinh monment
V d 10:
U (a, b)
M (t) =1
b a
Z ba
etxdx =ebt eat(b a) t .
Exp()
M (t) = Z +
0etx .exdx =
t (t < ) .
N; 2
M (t) = exp
t +
2t
2
P()
M (t) =
k=0etk
ek
k != exp ( exp (t) )
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 24 / 32
13. Hm sinh monment
Theorem
Nu hm sinh moment M (t) ca b.n.n tn ti trong ln cn ca 0
th n xc nh duy nht hm phn phi xc sut, ngha l khng thc hai bin ngu nhin khc nhau c cng hm sinh moment.
Nu v l hai bin ngu nhin c lp th
M+ (t) = M (t) .M (t)
V d 11: Gi s hm sinh moment ca mt bin ngu nhin X l
MX (t) = e3(et1).
Tnh P (X = 0).
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 25 / 32
14. S hi t ca cc bin ngu nhin14.1. Mt s bt ng thc quan trng
Lemma
(Bt ng thc Markov) Cho l bin ngu nhin khng m, ngha lP ( 0) = 1. Khi , vi mi c > 0,
P ( c) 1cE () .
Lemma
(Bt ng thc Chebyshev) Cho l bin ngu nhin c k vngE () = v phng sai Var () = 2. Khi , vi mi c > 0, ta c
P (j j c) 2
c2
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 26 / 32
14. S hi t ca cc bin ngu nhin14.2. Hi t theo xc sut
Ta ni dy bin ngu nhin (n) hi t theo xc sut n bin ngunhin nu vi mi > 0,
limn!
P (jn j > ) = 0.
K hiu nP! .
V d 12: Gi s Xn l bin ngu nhin c hm mt
fXn (x) =
8
14. S hi t ca cc bin ngu nhin14.3. Hi t theo phn b
Cho dy b.n.n (n) c cc hm phn b l Fn (x) v b.n.n c hm
phn b l F (x). Ta ni n hi t theo phn b v , k hiu
nd! , nu
limn!
Fn (x) = F (x) ,
ti mi x sao cho F lin tc.
V d 13: Gi s Xn = 1+1n l b.n.n hng s. Hm phn b ca Xn
l
FXn (x) =
0, x < 1+ 1/n,1, x 1+ 1/n.
Ta c Xnd! X , trong X = 1 l b.n.n hng s.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 28 / 32
14. S hi t ca cc bin ngu nhin14.4. Hi t trung bnh
Ta ni dy b.n.n (n) hi t theo trung bnh bc p v b.n.n , k hiu
nLp! , nu
limn!+
E (jn jp) = 0.
Quan h gia cc loi hi t
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 29 / 32
14. S hi t ca cc bin ngu nhin14.5. Lut s ln
Theorem (Lut s ln)
Gi s (n) l dy cc b.n.n c lp c cng phn b xc sut v
E (n) = ,Var (n) = 2, 8n 2 N.
Khi
n =1 + 2 + + n
n
P! khi n ! +.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 30 / 32
14. S hi t ca cc bin ngu nhin14.6. nh l gii hn trung tm
Theorem (nh l gii hn trung tm)
Gi s (n) l dy cc b.n.n c lp c cng phn b xc sut v
E (n) = ,Var (n) = 2, 8n 2 N.
Khi
n =(1 + 2 + + n) n
pn
d! Z N (0, 1) .
V d 14: Gi s n B (1, p) , 8n 2 N. Khi = 1 + 2 + + n B (n, p). Theo nh l gii hn trung tm, ta c
nppnp (1 p)
xp x vi b.n.n Z N (0, 1) .TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 31 / 32
14. S hi t ca cc bin ngu nhin14.6. nh l gii hn trung tm
V du 15: Gi s
- thi gian s dng mt loi pin c k vng 40 gi v phng sai 20 gi
- khi mt cc pin ht nng lng, ngi ta thay n bng mt cc khc
- ta s dng 25 cc pin v tui th ca chng l c lp
Tnh xc sut tng thi gian s dng ln hn 1100 gi.
TS. L Xun Trng (Institute) Tm tt L thuyt xc sut 32 / 32
Khng gian Xc sut