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Teorema 2 (liapunovis Teorema). Tu SemTxveviTi sidide warmoadgens
damoukidebel SemTxveviT sidideTa Zalian didi ricxvis jams, romelTaTvisac
Sesrulebulia piroba:
lim( n
n
bk)/[ Dk
] 0
n
k 1 k 1
,
sadac b k – mesame rigis absoluturi centraluri momentia
SemTxveviTi sididis , xolo D k – misi dispersia, maSin SemTxveviT
sidides gaaCnia ganawileba, romelic axlosaa normalur ganawilebasTan.
Tu SemTxveviTi sidide warmoadgens urTierTdamoukidebel SemTxveviT
sidideTa Zalian didi raodenobis jams, romelTagan TiToeulis gavlena
jamze mizerulad mcirea, maSin SemTxveviT sidides gaaCnia ganawileba,
romelic axlosaa normalurTan.
Teorema 3 (muavr-laplasis Teorema). Tu tardeba п damoukidebeli
cda, romelTagan TiToeulSi А xdomileba xdeba albaTobiT р, np 15 ,
maSin samarTliania Tanafardoba:
Sn
np
p
( ) ( ),
npq
sadac S n
– А xdomilebis moxdenaTa ricxvia п cdaSi, q = 1 – p, xolo
2
x t
1
() x
e
2
dt
2
Sedegi (muavr-laplasis lokaluri Teorema). muavr-laplasis Teoremis
pirobebSi р n
(k)
-- albaToba imisa, rom А xdomileba п cdaSi moxdeba
zustad k –jer, cdaTa didi ricxvis SemTxvevaSi, Tu np 15 , SeiZleba gamoiTvalos
Semdegi formuliT:
1
p n
( k)
(
x),
npq
x
k np
1
sadac x , xolo
2
( x)
e .
npq
2
magaliTi 1. dadgenilia, rom im adamianebis 94%-s, romelTac gakeTebuli
aqvT tuberkuliozis sawinaaRmdego acrebi, gamoumuSavdeba Sesabamisi
imuniteti. rogoria albaToba imisa, rom 100000 tuberkuliozze acrili
adamianidan 5800 araa daculi tuberkulioziT daavadebisagan?
amoxsna. am amocanis amoxsna marTalia Teoriulad SesaZlebelia bernulis
formuliT, sadac n 100000,
k 5800, p 0.06, q 0. 94, da Sesabamisad,
5800 5800 94200
P
100000
( 5800) C100000(0.06)
(0.94)
,
magram praqtikulad amis gakeTeba Zalian Znelia. am SemTxvevaSi unda visargebloT
muavr-laplasis lokaluri formuliT, romelic gvaZlevs:
5800100000
0.06
P
100000(5800)
(
)/ 100000
0.060.94
(
2.7)/75.
100000
0.060.94
85
2
3/2
k