Phan Phoi Xac Suat.pdf

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Cc phn phi xc sut thng gp

Nguyn Th Hng Nhung

Ngy 16 thng 11 nm 2013

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Table of contents

1 Cc phn phi ri rcPhn phi BernoulliPhn phi nh thcPhn phi Poisson

2 Cc phn phi lin tcPhn phi uPhn phi mPhn phi chun haPhn phi chunHm phn phi GammaPhn phi StudentPhn phi Chi bnh phng

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Bin ngu nhin Bernoulli

nh ngha 1 (Bin ngu nhin Bernoulli)

Thc hin mt php th, ta quan tm n bin c A. Nu binc A xy ra ( thnh cng) th X nhn gi tr l 1 (X = 1), ngcli bin ngu nhin X nhn gi tr 0. Php th ny gi l phpth Bernoulli. Gi s xc sut xy ra bin c A l p, 0 < p < 1

P(A) = P(X = 1) = p

vP(A) = P(X = 0) = 1 p = q.

khi , bin ngu nhin X c gi l bin ngu nhin c phnphi Bernoulli vi tham s p, k hiu X B(1; p).

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi Bernoulli

V d 1

Cc php th sau y cho kt qu l mt bin ngu nhinBernoulli

Tung ngu nhin mt ng xu: X = 1 nu xut hin mtsp, X = 0 nu xut hin mt nga.

Mua v s: X = 0 nu khng trng s, X = 1 nu trng s.

Tr li ngu nhin mt cu trc nghim : X = 0 nu tr ling, X = 1 nu tr li sai.

Nhn xt 1

Mi th nghim ngu nhin c hai kt qu u c phn phiBernoulli.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi Bernoulli

Bng phn phi xc sut

Bng phn phi xc sut ca bin ngu nhin X B(1; p) cdng

X 1 0

P p q

vi q = 1 p.Da vo bng phn phi xc sut ca bin ngu nhin X ta c

E(X ) = p

Var(X ) = pq.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi nh thc

nh ngha 2 (Binomial distribution)

Thc hin n php th Bernoulli c lp vi xc sut thnh cngtrong mi php th l p. Gi X l s ln thnh cng ( bin c Axy ra) trong n php th th

X = X1 + X2 + ...+ Xn

vi Xi , (i = 1, 2, ..., n), l bin ngu nhin c phn phi Bernoullivi cng tham s p.Khi X l bin ngu nhin ri rc vi min gi trS = {0, 1, ...., n} v xc sut

P(X = k) = C kn pkqnk , k S

X c gi l c phn phi nh thc vi tham s n, p k hiuX B(n, p).

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi nh thc- V d

V d 2

Mt l thuc, c t l hng p = 0.2 . Ta ly ngu nhin 5 l. GiX l s l hng trong s l ly ra. Tm hm mt xc sut caX .

Bi gii 1

Quan st l thuc trong 5 ln c lp. P(h) = p = .2.Gi X l s l b hng trong 5 l ly ra. Vy X {0, 1, 2, .., 5}v X B(5; 0.2) vi hm phn phi

f (h) =

{Ch5(0.2)

h(1 0.2)5h, h = 0, 1, ..., 50 ni khc

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi nh thc- V d

Ta c bng phn phi xc sut

H 0 1 2 3 4 5

P(H = h) 0.32768 .4096 .2048 0.0512 .0064 0.0032

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi nh thc- V d

V d 3

Trong mt nh my sn xut vi mch in t, bit rng t l vimch khng t cht lng l 5%. Kim tra ngu nhin 15 vimch. Tnh xc sut

a C ng 7 vi mch khng t cht lng

b C t nht 1 vi mch khng t cht lng

V d 4

Mt gia nh c 5 ngi con. Tnh xc sut gia nh ny

i C ng 2 con trai

ii c nhiu nht 2 con trai

iii c t nht 2 con trai.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi nh thc

nh l 1 (Cc c trng ca BNN c phn phi nh thc)

Nu X l bin ngu nhin c phn phi nh thc B(n; p) th

i E(X ) = np.ii Var(X ) = npq.iii Vi x , h l hai s nguyn dng th

P(x X x + h) = P(X = x) + P(X = x + 1) + ...... +P(X = x + h)

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi Poisson

nh ngha 3 (Poisson distribution)

Bin ngu nhin ri rc X nhn cc gi tr t 0, 1, 2, ... gi l cphn phi Poisson vi tham s , k hiu X P() nu hmmt xc sut c dng

f (x) = P(X = x) =ex

x!, x = 0, 1, 2, ...

vi > 0.

nh l 2 (Cc c trng ca bin ngu nhin c phn phiPoisson)

Nu bin ngu nhin X c phn phi Poisson vi tham s ,X P() th k vng v phng sai ca X ln lt bng

E(X ) = ,Var(X ) = .

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Xp x phn phi nh thc bng phn phi Poisson

nh l 3

Cho X B(n, p), nu n v p 0 sao cho np th

P(X = x) =ex

x!.

Trong thc t, phn phi Poisson s xp x tt cho phn phinh thc khi n 100 v np 20, p 0.01.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi Poisson

Mt s bin ngu nhin m t cc s kin sau thng cxem l tun theo phn phi Poisson

i S li in trong mt ( hoc mt s) trang sch.

ii S ngi sng lu trn 100 tui trong mt cng ng dnc.

iii S ngi n mt bu in no trong mt ngy.

iv S tai nn hoc s c giao thng xy ra ti mt im giaothng trong mt ngy....

Cc bin ngu nhin c s dng m t, " m" s ln xyra ca mt bin c, s kin no xy ra trong mt khong thigian v tha mt s iu kin ( cc iu kin ny thng thatrong thc t ) thng c m t bng phn phi Poisson.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi Poisson- V d

V d 5

Gi s s li in trong mt trang sch no ca quyn sch cphn phi Poisson vi tham s = 12 . Tnh xc sut c t nhtmt li in trong trang ny.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng


V d 6

S cuc in thoi gi n mt tng i in thoi trong mtgi c phn phi Poisson vi = 10. Tnh xc sut

i c 5 cuc in thoi gi n trong mt gi.

ii C nhiu nht 3 cuc in thoi gi n trong mt gi.

iii C 15 cuc gi n trong hai gi.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng


V d 7

Trong mt t tim chng cho tr em mt khu vc, bit xcsut mt tr b phn ng vi thuc sau khi tim l 0, 001. Thchin tim cho 2000 tr. Tnh xc sut c nhiu nht 1 tr bphn ng vi thuc sau khi tim.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi u

nh ngha 4 (Uniform distribution)

Bin ngu nhin lin tc X c gi l c phn phi u trnon [a; b], k hiu X U([a; b]), nu hm mt xc sut caX c dng

f (x) =1

b a , khi x [a; b]

v

f (x) = 0,x / [a; b]T nh ngha trn ta c th tnh c hm phn phi xc

sut ca X U([a; b]) nh saux < a,F (x) = 0x [a, b],F (x) = xaba ,x > b,F (x) = 1.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

K vng v phng sai ca phn phi u

nh l 4 (Cc c trng ca bin ngu nhin c phn phi u)

Cho X l bin ngu nhin c phn phi u trn [a, b] ( tc lX U([a, b])) th

i K vng E(X ) = a+b2 .

ii Phng sai Var(X ) = (ba)2

12 .

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi m

nh ngha 5 (Exponential distribution)

Bin ngu nhin T (t > 0) gi l c phn phi m, k hiuT Exp(), nu n c hm mt xc sut

f (t) = et , t > 0,

trong

: s bin c trung bnh xy ra trong mt n v thi gian.

t: s n v thi gian cho bin c k tip.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Cc c trng ca phn phi m

Hm phn phi ca T :

F (t) = P(T t) = 1 et , t > 0.

nh l 5

Nu T Exp() th k vng v phng sai ca T ln lt l

E(T ) =1

Var(T ) =1

2

nh l 6

Nu T l bin ngu nhin c phn phi m th,

P(T < t1 + t2|T > t1) = P(T < t2)

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun ha

nh ngha 6

Cho bin ngu nhin X lin tc, X c phn phi chun ha, khiu X N (0, 1), khi hm mt c dng sau

f (x) =12pi

ex22 .

nh l 7

Bin ngu nhin X N (0, 1) c k vng E(X ) = 0 v phngsai Var(X ) = 1.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun ha-hm mt

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun ha-hm phn phi

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun ha

nh l 8 (Hm phn phi)

(x) = P(X < x) = x

12pi

et22 dt.

Vi gi tr c th ca x , ta tra bng tm gi tr (x).

V d 8

Cho bin ngu nhin X N (0, 1). Tnh cc xc sut sauP(X < 1.55)P(X < 1.45)P(1 X < 1.5)

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun

nh ngha 7 (Normal distribution)

Bin ngu nhin lin tc X nhn gi tr trong khong(,+) c gi l c phn phi chun vi tham s , nuhm mt xc sut c dng

f (x) =1

2pi

exp

((x )

2

22

), < x 0, <

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun- Tnh cht

* th c dng hnh chung

* Phn phi i xng

* Trung bnh = trung v (median)= Mode

* V tr ca phn phi c xc nh bi k vng

* phn tn c xc nh bi lch tiu chun

* Xc nh trn R.

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun- Hm mt

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun- Hm phn phi

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi chun

nh l 9

Nu X N (, 2) th X ,.

H qu 1

Nu X N (, 2) th

P(X < a) = (a

).

H qu 2

Nu X N (, 2) th

P(a X < b) = (b

)

(a

).

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Hm phn phi Gamma

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi Student

Cc phnphi xcsut

thng gp

NguynTh HngNhung

Cc phnphi ri rc

Phn phiBernoulli

Phn phi nhthc

Phn phiPoisson

Cc phnphi lintc

Phn phi u

Phn phi m

Phn phi chunha

Phn phi chun

Hm phn phiGamma

Phn phiStudent

Phn phi Chibnh phng

Phn phi Chi bnh phng

Cc phn phi ri rac Phn phi BernoulliPhn phi nhi thcPhn phi Poisson

Cc phn phi lin tucPhn phi uPhn phi muPhn phi chun haPhn phi chunHm phn phi GammaPhn phi StudentPhn phi Chi bnh phng

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Phan Phoi Xac Suat.pdf