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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
Phn phi Poisson- V d
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
Phn phi Poisson- V d
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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