[PDF] Normal approximation to the binomial




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[PDF] The Normal Approximation to the Binomial Distribution

Remarkably, when n, np and nq are large, then the binomial distribution is well approximated by the normal distribution According to eq (8 3) on p 762 of Boas 

[PDF] Normal approximation to the binomial

Theorem 9 1 (Normal approximation to the binomial distribution) If Sn is a binomial variable with parameters n and p, Binom (n, p), then

[PDF] Approximating the Binomial Distribution by the Normal - DiVA Portal

21 jui 2011 · In this paper an examination is made regarding the size of the approximations errors The exact probabilities of the binomial distribution is 

[PDF] Lab Project 5: The Normal approximation to Binomial distribution

Lab Project 5: The Normal approximation to Binomial distribution Course : Introduction to Probability and Statistics, Math 113 Section 3234

[PDF] 65 The Normal Approximation to the Binomial Distribution

12 nov 2019 · Although it seems strange, under certain circumstances a (continuous) normal distribution can be used to approximate a (discrete) binomial 

[PDF] Normal approximation to the Binomial

In 1733, Abraham de Moivre presented an approximation to the Binomial distribution He later (de Moivre, 1756, page 242) appended the derivation

[PDF] Normal Distribution as Approximation to Binomial Distribution

Normal Approximation to the Binomial distribution IF np > 5 AND nq > 5, then the binomial random variable is approximately normally distributed with mean µ =np 

[PDF] NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

For accurate values for binomial probabilities, either use computer software to do exact calculations or if n is not very large, the probability calculation 

[PDF] Section 55, Normal Approximations to Binomial Distributions

Let x represent a binomial random variable for n trials, with probability of Since the binomial distribution is discrete and the normal distribution is 

[PDF] The Normal Distribution as an Approximation to the Binomial

For a large enough number of trials (n) the area under normal curve can be used to approximate the probability of a binomial distribution Requirements:

[PDF] Normal approximation to the binomial 29623_6prob3160ch9.pdf ??????? ? ?????? ????????????? ?? ??? ????????

? ??????? ???? ?? ?????????? ????? ????????? ??? ????????? ????????????????? ??? ??????? ????????????? ?? ??? ???????? ?????????????

??Sn?? ? ???????? ???????? ???? ??????????n???p?Binom(n;p)? ???? P a6Snnppnp(1p)6b! ! n!1P(a6Z6b);

??n! 1? ?????Z N(0;1)????? ????????????? ?? ???? ??np(1p)>10??? ???? ?????? ??? ?????? ???? ???????? ?????

???? ????? ???? ?? ??????p??1p?? ?????? ???? ???? ?? ????? ??? ?????n? ?????? ???? ?? ??????????? ???np?? ??? ???? ??ESn???np(1p)?? ??? ???? ??VarSn? ?? ??? ????? ?? ????? ??(SnESn)=pVarSn? ??? ???? ????? ??? ???? ? ??? ???????? ?? ??? ???? ?? ? ????????N(0;1)? ???? ???? ????p????? ???? ??n! 1? ?????? ?? ??? ???? ?? ??? ??????? ?????????????? ?? ?? ????????? ?? ??????????? ???? ?????? ?? ??? ?????????? ?? ??????? ??? ??????? ?? ???? ??????? ?? ?? ???? ??????? ??? ??? ????? ???? ?? ???? ??? ???? ?????? ?? ????? ???? ?? ??????? ??? ???????????? ?? S n???? ??? ???????????? ?? ??? ?????? ????????X N np;pnp(1p) ? ??? ?????? ????????X??? ??? ???????

1p2np(1p)e(xnp)22np(1p):

??? ???? ?????? ???? ????? ?? ???? ?? ??? ?????????? ?? ????????????? ??? ???????? ???? ????????? ?? ??? ?????? ???????????? ?? ??? ?????? ????? ??? ???????? ???????????? ?????? ???????????? ???? ????? ???? ??? ?? ??? ?????? ?????? ??? ????np? ?? ????????P(Sn=k)? ??? ?? ?????? ????k???? ??? ??????? ??? ???? ????np? ?? ??????? ?????????? ?? ???? ????? ?????? ?? ???????? ??????????? ????? ??pnp(1p)? ????????? ?? ??? ????knp ?????? ?? ?? ?????pn? ???? ?? ??? ???? ?? ? ??????????? ????? ????k???????? ????np?? ???? ???????? ???????????P(Sn=k)??????? ???? ????? ??? ??? ?? ???????????? ?? ????? ?? ???? ??????? ?? ?????? ????k???nk?? ?????n? ?? ??? ?????????? ??????? ?? ??? ????????? ???? m!p2memmm; ??? ??? ?? ?????? ????????????? ????? ???? ???? ???? ??? ??? ?????????? ??? ?????????????? ?????? ???? ???????? ????? ????? ??1??m! 1? ???? ??? ?????n?k???nk

P(Sn=k) =n!k!(nk)!pk(1p)nk

 p2nennnp2kekkkp2(nk)e(nk)(nk)nkpk(1p)nk = pk  k1pnk nk n nrn

2k(nk)=npk

 kn(1p)nk nkrn

2k(nk):

??? ?? ??? ??? ?????????? ln npk  =ln

1 +knpnp

 ; ln n(1p)nk =ln

1knpn(1p)

: ???? ?? ??? ???ln(1 +y)yy22 +y33 ;y!0?? ??? ???? ln npk  kn(1p)nk nk! =klnnpk  + (nk)lnn(1p)nk k knpnp +12  knpnp  2 13  knpnp  3! + (nk) knpn(1p)+12  knpn(1p) 2 +13  knpn(1p) 3!  (knp)22np(1p): ???? npk  kn(1p)nk nk e(knp)22np(1p): ??? ?? ??? ??? ?????????? ????knp?????? ?? ?? ?????pn?? ??? ???? knppn; nkn(1p)pn; k(nk)n2p(1p); ?? ?? ?????? ????????????? ??? rn

2k(nk)1p2np(1p):

 ??????? ??????????? ? ???? ???? ?? ?????? ??? ?????? ???? ?? ??? ??????????? ????? ???? ?? ???? ????60?????? ?????????np= 50???pnp(1p) = 5? ?? ????

P(Sn>60) =P((Sn50)=5>2)P(Z>2)0:0228:

??????? ??????????? ? ??? ?? ?????? ??? ?????? ???? ?? ??? ??????????? ? ? ???? ?? ??????? ???? ???? ?? ?????? ????????? ????p=16 ? ??np= 30???pnp(1p) = 5? ????P(Sn>50)P(Z >4)? ????? ?? ???? ????e42=2? ??????? ??????????? ? ???? ?? ???????? ?? ?? ??? ????????? ?? ?? ?????? ?? ??? ??????? ???? ?? ??? ??????????? ???? ???? ?? ?????? ???? ?? ??????? ????????? ????Sn?? ??? ?????? ?? ??????????n= 100? ???p= 0:75? ?? ????

P(Sn>70) =P((Sn75)=p300=16>1:154)

P(Z>1:154)0:87: ???? ???? ????? ???? ???? ? ??????? ????ba?? ?????? ????? ?? ? ?????????? ???? ????? ?????? ???? ????????? ?????? ??????? a??a12 ???b??b+12 ? ???? ?????????? ????? ????? ??? ?? ???????? ?????????? ??? ???????? ?? ??????? ? ???? ??? ?????? ????? ?? ???????? ??????????? ???? ????? ??? ??????? ?? ?????? ????? ??????? ??? ??????????? ??? ?????? ????? ?? ??? ?????? ????????????? ????? ?? ?? ??????? ?????? ???? ? ???? ??? ?????? ???? ?? ??? ??????????? ?? ??????? ??? ??? ?? ?? ?????? ????????? ?? ?????P(496Sn651) =P(48:56Sn651:5)??? ???? ???????? ?? ?????? ??? ?? ?????? ????????????? ?? ???? ???? ?? ????? ???? p= 0:5; =np= 50; 

2=np(1p) = 25;

=pnp(1p) = 5: ??? ?????? ????????????? ??? ?? ???? ?? ????? ???????? ????? P(496Sn651)P(49650 + 5Z651) = (0:2)(0:2) = 2(0:2)10:15852 ?? P(48< Sn<52)P(48<50 + 5Z <52) = (0:4)(0:4) = 2(0:4)10:31084 ?? P(48:5< Sn<51:5)P(48:5<50+5Z <51:5) = (0:3)(0:3) = 2(0:3)10:23582 ???? ??? ????? ??????? ??? ???????????? ??? ??? ????? ???????????? ?? ??? ???? ???????? ????? ????? ?????? ?? ???? ??? ??????? ??? ??????? ?????? ????? ??? ???????? ????????

P(496Sn651) =51X

k=49 100
k12  100
=37339688790147532337148742857158456325028528675187087900672 0:2356465655973331958::: ?? ???????? ?? ??? ?????? ??? ????????? ?????? ?????????????? P(Sn= 49)P(48:5650 + 5Z649:5) = (0:1)(0:3) = (0:3)(0:1)0:07808 P(Sn= 50)P(49:5650 + 5Z650:5) = (0:1)(0:1) = 2(0:1)10:07966 P(Sn= 51)P(50:5650 + 5Z651:5) = (0:3)(0:1)0:07808 ???????? ?????? ????

0:07808 + 0:07966 + 0:07808 = 0:23582

????? ?? ??? ??????????? ????? ???P(496Sn651)P(48:5<50 + 5Z <51:5)??????????? ?????????? ?? ? ?????????? ???????????? ???? ?? ??? ?????? ???????????? ?? ???? ?? ??????????? ?

???????? ??? ???? ?? ??? ???????? ????????????? ??????????? ???????????????? ?? ???????? ???????? ??X?? ? ???????? ?????? ???????? ???? ?????????? ??? ?????? ?? ????????? ??

n??????????? ?????? ???? ??? ??????????? ?? ??????? ?? ??? ?????p? ???Y?? ? ?????? ?????? ???????? ???? ??? ???? ???? ??? ??? ???? ???????? ??X? ???? ??? ??? ???????k?? ???? ????P(X6k)?? ???? ???????????? ??P(Y6k)??np(1p)?? ??? ??? ?????? ?? ?? ?????? ???????????? ??P(Y6k+ 1=2)?? ????????? ?? ??? ??? ?? ???? ???????? ??? ???? ??1=2 ?? ????? ?? ?? ????? ?? ??????? ?? ??? ?????? ???????????? ????? ??? ?????? ??? ?? ??? ?? ?? ??????????? ??? ???????? ????????????? ?? ?????? ????????????? ??? ??? ???? ???? ???? ????????????? ?? ?????? ????? ?? ? ?????? ?? ??????????????? ??? ?? ???? ? ???????? ?????? ???????? ??? ???? ???? ?? ???? ???????? ?????? ???? ?? ????????? ????? ? ?????????? ?????? ???????? ???? ?? ??????????? ?? ??? ???? ?? ??? ?????? ?????? ?? ???????? ?????? ????? ???????? ??????? ?????? ???? ????? ??? ?????? ???????????? ?? ??????????? ??? ????????? ???? ???????? ?????????????? ??? ?????? ?? ?? ???????? ?? ? ?????????? ?????????? ?? ????? ??? ?????? ?????? ?? ???? ? ?????????? ???????????? ???? ?? ??? ??????? ??? ??????????? ?? ?????? ?? ? ?????????? ????? ?? ? ?????? ???????? ?? ????? ?? ??? ????? ????? ???? ??? ?????? ????????????? ?? ???? ?? ??????????? ? ???????? ????????????? ? ?????????? ?????????? ??? ?? ???????? ?? ???? ?? ??? ??????????? ??? ??????????? ?? ? ??????? ????? ?? ??? ???????? ????????????? ??? ???????? ?? ?? ???? ?? ???????????P(36X65) =P(X= 3??X= 4??X= 5)?? ? ?????? ????????????? ?? ????? ?? ????????????????? ?? ???P(Y= 3??Y= 4??Y= 5) ?? ??? ??????????? ??Y?????? ??3?4???5??0? ?? ??? ????????????? ???????????? ??? ????

P(36X65) =P(2:56X65:5)

?????????? ??? ?????? ????????????? ??P(2:56Y65:5)? ????? ?? ? ????? ?? ??? ?? ??? ??? ?????????? ?????????? ??? ?????? ????????????? ?? ? ???????????????????????

??P(X=n)???P(n0:5< X < n+ 0:5)??P(X > n)???P(X > n+ 0:5)??P(X6n)???P(X < n+ 0:5)??P(X < n)???P(X < n0:5)??P(X>n)???P(X > n0:5)

??? ?? ?????? ????????????? ???? ????????? ???????? ??????????? ???? ?? ???? ? ???? ??? ?????? ???E?? ??? ????? ???? ?? ???? ??? ???? ?? ???? ???? ????? ??? ???? ??? ????? ?????????? ??? E? ??? ????? ??????P(E)????? ??? ?????? ????????????? ??? ????? ??????P(E)????? ??? ??????? ????????????? ???????? ?????????10%?? ??? ?????????? ?? ???????????? ??? ??? ?????? ???????????? ?? ??????????? ??? ??????????? ???? ?? ? ????? ?? ??? ????????? ??? ?? ????? ?? ?? ???? ??? ???????????? ??? ? ???????? ??? ?? ??? ???????????? ???????? ????? ??????? ????????? ? ??? ???? ?? ??????? ?? ?????? ???????? ?? ??????? ??? ??????????? ???? ?????? ?? ????? ?? ???? ??????????? ?? ??? ????? ???????? ??????? ???? ??? ??????????? ?? ????? ????? ???? ???? ? ????? ??????? ?? ??????? ???? ??? ??? ??? ??????????? ?? ????? ????? ???? ???? ? ? ? ????? ??????? ?? ???? ???? ???? ??? ??? ?????? ????????????? ?? ????????? ?? ???? ???????? ????????? ??? ???? ???????? ????????? ???????? ?? ???????? ?????????? ??????????? ?? ??????? ??? ???? ?? ? ?????????? ???? ??136 ? ?? ??? ??????????? ???? ?? ???? ??? ???? ?? ???? ???? ???? ?? ??? ????? ?? p=0 B B@180 01 C CA 3536
 180
+0 B B@180 11 C CA 136
 3536
 179
:0386: ???????? ?? ???????? ????????? ??? ?????????? ?? ??? ?????? ?? ????????? ?? ?? ? ?? ?? ???? ???P(06S18061)? ????? ??? ???????? ?? ??????????????? ?? ????? ??? ?????????? ?????????? ??? ?????????P(0:56S18061:5)???????? ?? ??? ???? ??? ???????? ????? ?? = 180p= 5??? ??? ???????? ????????? ??=p180p(1p)2:205? ???? ?? ??????? ?? ??????? ???? ???????? ?? ? ???????? ????? ??? ???????? ?????? ?????? ????????Z?

P(0:56S18061:5) =P0:552:2056Z61:552:205

=P(2:49< Z <1:59) = (1(1:59))(1(2:49)) = (10:9441)(10:9936) = 0:0495: ???????? ?? ???????? ????????? ???=np= 5????? ???? ?? ????? ???? ??????? ?? ?????? ??? ?? ??? ????

P(E)e5500!

+e5511! 0:0404: ???????? ?? ???????? ???????X?????? ??? ?????? ?? ??????????? ???????? ?? ??? ?????? ?? ??? ??????? ??? ????XBinom(150;0:1)?????? ???? ????np= 15? ???P(X>25) =P

X15p13:5>10p13:5

1(2:72)0:00364? ???? ???? ?? ???? ????????????? ?? ?????????? ??? ???? ??? ???? ?? ???? ??????????? ???????????? ?? ?????? ??? ?? ??????? ?? ? ????????? ???????? ?? ???? ???? ? ????????? ????????? ?? ??? ??? ???? ??? ?????? ?? ?????? ????? ?? ??? ????????? ????????? ?? ????????

P(150>X>25) =P135p13:5>X15p13:5>10p13:5

(36:74)(2:72)0:00364: ???????? ???? ????????????? ????????????? ???????? ??

P(150>X>25) =P(150:5>X>24:5)

=P135:5p13:5>X15p13:5>9:5p13:5 (36:87)(2:59)0:00480: ??? ????????? ?? ??? ???? ???? ??? ?? ?????? ?????????????

P(156X620) =P(14:5< X <20:5)

= 5:5p13:5 0:5p13:5 (1:5)1 + (0:14)0:4889: ???????? ?? ???????? ???????X?????? ??? ?????? ?? ????? ???????? ?????X

Binom(50;0:6)?? ????

P(X > N)1N302

p3  >0:2???P(X > N+ 1)1N292 p3  <0:2: ???? ???? ?? ?????? ????N632:909???N>31:944?? ????N= 32?
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