Għal suċċessjoni ta' varjabbli każwali
X
1
,
X
2
,
.
.
.
,
X
n
,
.
.
.
{\displaystyle \ X_{1},X_{2},...,X_{n},...}
indipendenti u distribwiti identikament b'medja
μ
{\displaystyle \ \mu }
, il-medja kampjunarja hi
X
¯
n
=
X
1
+
X
2
+
⋯
+
X
n
n
.
{\displaystyle {\bar {X}}_{n}={{X_{1}+X_{2}+\cdots +X_{n}} \over {n}}.}
Il-liġi (qawwija) tan-numri kbar tgħid li
P
(
lim
n
→
∞
X
¯
n
=
μ
)
=
1
,
{\displaystyle \operatorname {P} \left(\lim _{n\rightarrow \infty }{\bar {X}}_{n}=\mu \right)=1,}
jiġifieri, il-medja kampjunarja tikkonverġi kważi ċertament għall-medja komuni tal-
X
i
{\displaystyle \ X_{i}}
.
Il-liġi (dgħajfa) tan-numri kbar tgħid li jekk
X
1
,
X
2
,
.
.
.
,
X
n
,
.
.
.
{\displaystyle X_{1},\,X_{2},\,...,\,X_{n},\,...}
tkun suċċessjoni ta' varjabbli każwali li għandhom l-istess medja
μ
{\displaystyle \ \mu }
, l-istess varjanza finita u indipendenti, imbagħad
għal kull
ε
>
0
{\displaystyle \ \varepsilon >0}
:
lim
n
→
∞
P
(
|
X
¯
n
−
μ
|
<
ε
)
=
1
,
{\displaystyle \lim _{n\rightarrow \infty }\operatorname {P} \left(\left|{\bar {X}}_{n}-\mu \right|<\varepsilon \right)=1,}
jiġifieri, il-medja kampjunarja tikkonverġi fil-probabbiltà għall-medja komuni tal-
X
i
{\displaystyle \ X_{i}}
.
Il-liġi tan-numri kbar tiggarantixxi li l-medja kampjunarja tagħtina stima konsistenti tal-medja ta' popolazzjoni; biżżejjed ngħidu li mħabba l-liġi tan-numri kbar nistgħu ikkolna fiduċja li l-medja li nikkalkulaw minn numru kbir biżżejjed ta' kampjuni hi qrib biżżejjed tal-medja vera.
Nissoponu li għandna ġrajja (bħall-fatt li t-tfigħ ta' damma jagħtina s-sitta) b'probabbiltà li ma nafuhiex
p
{\displaystyle \ p}
(ma nafuhiex għax id-damma tista' tkun imbabsa, jew sempliċement difettuża: ma nistgħux inkunu nafu minn qabel).
Jekk nitfgħu id-damma
n
{\displaystyle \ n}
darba wara xulxin niksbu stima tal-probabbiltà li nġibu s-sitta b'dik id-damma,
p
{\displaystyle \ p}
, li hi mogħtija minn
X
¯
n
=
X
1
+
X
2
+
.
.
.
+
X
n
n
{\displaystyle {\bar {X}}_{n}={\frac {X_{1}+X_{2}+...+X_{n}}{n}}}
fejn kull
X
{\displaystyle \ X}
fis-somma tirrappreżenta tefgħa u tiswa wieħed jekk it-tefgħa tagħtina s-sitta u żero jekk jiġi numru ieħor. Il-liġi tan-numri kbar tafferma sempliċement li, iżjed ma nużaw provi biex nikkalkulaw l-istima, iżjed din tkun qrib, probabbilment , għall-probabbiltà vera tal-ġrajja,
p
{\displaystyle \ p}
.
Jekk l-istima
X
(
n
)
{\displaystyle \ X(n)}
li nikkalkulaw tkun qrib ħafna ta' wieħed f'sitta, li hi l-probabbiltà teorika li nġibu s-sitta għall-damma perfetta, nistgħu inkun ċerti mhux ħażin li d-damma m'hijiex imxaqilba lejn is-sitta (biex inkunu żguri li d-damma ma xxaqlibx lejn l-ebda numru irridu nirrepetu l-provi għall-ħames numri l-oħra). Xi tfisser żguri mhux ħażin jiddipendi minn kemm irridu nkunu preċiżi fil-provi tagħna: b'għaxar tefgħat ikollna stima raffa, b'mija jkollna waħda iżjed preċiża, b'elf iżjed u nibqgħu sejjrin hekk: il-valur ta'
n
{\displaystyle \ n}
li lesti li naċċettaw bħala biżżejjed jiddependi mill-grad ta' każwalità li naħsbu li hu neċessarju għad-damma li qegħdin nużaw.
Ħalli
{
(
Ω
i
,
A
i
,
P
i
)
}
i
∈
N
{\displaystyle \{(\Omega _{i},{\mathcal {A}}_{i},\operatorname {P} _{i})\}_{i\in \mathbb {N} }}
tkun suċċessjoni ta' spazji ta' probabbiltà . Inħarsu lejn l-ispazju prodott
(
Ω
,
A
,
P
)
{\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}
u fih suċċessjoni Bernoulljana ta' ġrajjiet (stokastikament indipendenti u b'probabbiltà kostanti
p
{\displaystyle \ p}
),
{
E
k
}
k
∈
N
⊂
A
{\displaystyle \{E_{k}\}_{k\in \mathbb {N} }\subset {\mathcal {A}}}
. Għal kull element
ω
∈
Ω
{\displaystyle \omega \in \Omega }
niddefinixxu l-frekwenza ta' suċċess f'
n
{\displaystyle \ n}
provi,
ϕ
n
:
Ω
→
R
,
ϕ
n
(
ω
)
=
N
n
/
n
{\displaystyle \phi _{n}:\Omega \to \mathbb {R} ,\phi _{n}(\omega )=N_{n}/n}
, fejn
N
n
=
#
{
i
:
ω
∈
E
i
}
i
=
1
n
{\displaystyle N_{n}=\#\{i:\omega \in E_{i}\}_{i=1}^{n}}
turi n-numru ta' suċċessi miksuba f'
n
{\displaystyle \ n}
provi.
B'din in-notazzjoni il-liġi nistgħu niktbuha:
∀
ε
>
0
{\displaystyle \forall \ \varepsilon >0}
,
lim
n
→
∞
P
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
=
0
{\displaystyle \lim _{n\to \infty }\operatorname {P} \{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \}=0}
.
Prova:
Jekk niffissaw
ε
{\displaystyle \ \varepsilon }
u nużaw id-diżugwaljanza ta' Čebyšëv [ 2]
P
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
E
(
ϕ
n
)
|
>
ε
}
≤
var
(
ϕ
n
)
ε
2
{\displaystyle \operatorname {P} \{\omega \in \Omega :|\phi _{n}(\omega )-\operatorname {E} (\phi _{n})|>\varepsilon \}\leq {\frac {\operatorname {var} (\phi _{n})}{\varepsilon ^{2}}}}
Jekk
N
n
{\displaystyle \ N_{n}}
għandha distribuzzjoni binomjali , jkollna
E
(
N
n
)
=
p
n
{\displaystyle \operatorname {E} (N_{n})=pn}
u
var
(
N
n
)
=
n
p
(
1
−
p
)
,
{\displaystyle \operatorname {var} (N_{n})=np(1-p),}
mil-liema
E
(
ϕ
n
)
=
p
{\displaystyle \operatorname {E} (\phi _{n})=p}
u
var
(
ϕ
n
)
=
1
n
2
n
p
(
1
−
p
)
=
p
(
1
−
p
)
n
{\displaystyle \operatorname {var} (\phi _{n})={\frac {1}{n^{2}}}np(1-p)={\frac {p(1-p)}{n}}}
.
Meta nissostitwixxu niksbu:
P
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
≤
p
(
p
−
1
)
n
ε
2
{\displaystyle \operatorname {P} \{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \}\leq {\frac {p(p-1)}{n\varepsilon ^{2}}}}
u la
lim
n
→
∞
p
(
p
−
1
)
n
ε
2
=
0
{\displaystyle \lim _{n\to \infty }{\frac {p(p-1)}{n\varepsilon ^{2}}}=0}
,
∀
ε
>
0
{\displaystyle \forall \ \varepsilon >0}
,
lim
n
→
∞
P
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
≤
0
{\displaystyle \lim _{n\to \infty }\operatorname {P} \{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \}\leq 0}
Imma
P
(
A
)
≥
0
{\displaystyle \operatorname {P} ({\mathcal {A}})\geq 0}
, u għalhekk ipprovajna l-liġi dgħajfa.
Nota: Il-liġi dgħajfa tan-numri kbar ma tiżgurax li, nagħżlu kif nagħżlu
ε
>
0
{\displaystyle \varepsilon >0}
, kważi ċertament jekk nibdew minn ċertu
n
ε
{\displaystyle n_{\varepsilon }}
il-valur ta'
|
ϕ
n
−
p
|
{\displaystyle \ |\phi _{n}-p|}
ħa jibqa inqas jew daqs
ε
{\displaystyle \varepsilon }
, jiġifieri li s-sett
{
ω
∈
Ω
:
∃
n
ε
:
∀
n
>
n
ε
,
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
{\displaystyle \{\omega \in \Omega :\exists \ n_{\varepsilon }:\forall \ n>n_{\varepsilon },|\phi _{n}(\omega )-p|>\varepsilon \}}
ħa jkun
P
{\displaystyle \operatorname {P} }
-traskurabbli.
Infatti, jekk nagħmlu d-definizzjoni tal-limitu iżjed espliċita, insibu li:
∀
ε
>
0
,
∀
η
>
0
,
∃
n
ε
,
η
:
∀
n
≥
n
ε
,
η
,
P
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
≤
η
{\displaystyle \forall \ \varepsilon >0,\forall \ \eta >0,\exists \ n_{\varepsilon ,\eta }:\forall \ n\geq n_{\varepsilon ,\eta },\operatorname {P} \{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \}\leq \eta }
imma m'hemm xejn li jiżgura li
n
ε
,
η
{\displaystyle n_{\varepsilon ,\eta }}
ma tiddiverġiex meta
η
→
0
{\displaystyle \eta \to 0}
.
Minn naħa l-oħra l-liġi qawwija tan-numri kbar: :
P
{
ω
∈
Ω
:
lim
n
→
∞
ϕ
n
(
ω
)
=
p
}
=
1
{\displaystyle \operatorname {P} \{\omega \in \Omega :\lim _{n\to \infty }\phi _{n}(\omega )=p\}=1}
timplika li
∀
ε
>
0
{\displaystyle \forall \ \varepsilon >0}
,
P
{
ω
∈
Ω
:
∃
n
ε
:
∀
n
>
n
ε
,
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
=
0.
{\displaystyle \operatorname {P} \{\omega \in \Omega :\exists \ n_{\varepsilon }:\forall \ n>n_{\varepsilon },|\phi _{n}(\omega )-p|>\varepsilon \}=0.}
u din l-asserzjoni tal-aħħar timplika l-liġi dagħjfa tan-numri kbar.
Prova taż-żewġ implikazzjonijiet:
1.
Billi nagħmlu d-definizzjoni tal-limitu espiliċita u ngħaddu għall-kumplement, nistgħu nifformolaw il-liġi qawwija b'dal-mod:
P
{
ω
∈
Ω
:
∃
ε
>
0
:
∀
n
ε
∈
N
,
∃
n
>
n
ε
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
=
0.
{\displaystyle \operatorname {P} \{\omega \in \Omega :\exists \ \varepsilon >0:\forall \ n_{\varepsilon }\in \mathbb {N} ,\exists \ n>n_{\varepsilon }:|\phi _{n}(\omega )-p|>\varepsilon \}=0.}
Meta nittrasformaw il-kwantifikatur eżistenzjali f'unjoni, din issir:
P
(
⋃
ε
>
0
{
ω
∈
Ω
:
∀
n
ε
∈
N
,
∃
n
>
n
ε
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
)
=
0.
{\displaystyle \operatorname {P} (\bigcup _{\varepsilon >0}\{\omega \in \Omega :\forall \ n_{\varepsilon }\in \mathbb {N} ,\exists \ n>n_{\varepsilon }:|\phi _{n}(\omega )-p|>\varepsilon \})=0.}
Issa jekk
ε
>
0
{\displaystyle \varepsilon >0}
, bil-monotonija ta'
P
{\displaystyle \operatorname {P} }
għandna
0
≤
P
{
ω
∈
Ω
:
∃
n
ε
∈
N
:
∀
n
>
n
ε
,
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
{\displaystyle 0\leq \operatorname {P} \{\omega \in \Omega :\exists \ n_{\varepsilon }\in \mathbb {N} :\forall \ n>n_{\varepsilon },|\phi _{n}(\omega )-p|>\varepsilon \}}
≤
P
(
⋃
ε
>
0
{
ω
∈
Ω
:
∀
n
ε
∈
N
,
:
∃
n
>
n
ε
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
)
=
0
,
{\displaystyle \leq \operatorname {P} (\bigcup _{\varepsilon >0}\{\omega \in \Omega :\forall \ n_{\varepsilon }\in \mathbb {N} ,:\exists \ n>n_{\varepsilon }:|\phi _{n}(\omega )-p|>\varepsilon \})=0,}
u mela
P
{
ω
∈
Ω
:
∃
n
ε
∈
N
:
∀
n
>
n
ε
,
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
=
0.
{\displaystyle \operatorname {P} \{\omega \in \Omega :\exists \ n_{\varepsilon }\in \mathbb {N} :\forall \ n>n_{\varepsilon },|\phi _{n}(\omega )-p|>\varepsilon \}=0.}
2.
Minn naħa l-oħra jekk nassumu din tal-aħħar u nittrasformaw ukoll il-kwantifikaturi f'operazzjoniet tas-settijiet, ikollna:
0
=
P
{
ω
∈
Ω
:
∃
n
ε
∈
N
:
∀
n
>
n
ε
,
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
{\displaystyle 0=\operatorname {P} \{\omega \in \Omega :\exists \ n_{\varepsilon }\in \mathbb {N} :\forall \ n>n_{\varepsilon },|\phi _{n}(\omega )-p|>\varepsilon \}}
=
P
(
⋂
m
∈
N
⋃
n
>
m
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
)
{\displaystyle =\operatorname {P} (\bigcap _{m\in \mathbb {N} }\bigcup _{n>m}\{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \})}
Imma, billi hemm intersezzjoni ta' suċċessjoni ta' settijiet li ma jikbrux, bil-monotonija ta'
P
{\displaystyle \operatorname {P} }
, nistgħu niktbu:
lim
m
→
∞
P
(
⋃
n
>
m
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
)
=
0.
{\displaystyle \lim _{m\to \infty }\operatorname {P} (\bigcup _{n>m}\{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \})=0.}
Imbagħad bil-monotonija ta'
P
{\displaystyle \operatorname {P} }
niksbu għal kull
ε
>
0
{\displaystyle \varepsilon >0}
.
lim
n
→
∞
P
(
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
)
≤
lim
m
→
∞
P
(
⋃
n
>
m
{
ω
∈
Ω
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
)
=
0
{\displaystyle \lim _{n\to \infty }\operatorname {P} (\{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \})\leq \lim _{m\to \infty }\operatorname {P} (\bigcup _{n>m}\{\omega \in \Omega :|\phi _{n}(\omega )-p|>\varepsilon \})=0}
li hi l-liġi dagħjfa tan-numri kbar.
Prova tal-liġi qawwija:
Digà rajna li l-asserzjoni hi ekwivalenti għal:
P
(
⋃
ε
>
0
{
ω
∈
Ω
:
∀
n
ε
∈
N
,
∃
n
>
n
ε
:
|
ϕ
n
(
ω
)
−
p
|
>
ε
}
)
=
0
{\displaystyle \operatorname {P} (\bigcup _{\varepsilon >0}\{\omega \in \Omega :\forall \ n_{\varepsilon }\in \mathbb {N} ,\exists \ n>n_{\varepsilon }:|\phi _{n}(\omega )-p|>\varepsilon \})=0}
Din hi wkoll ekwivalenti għal:
P
(
⋃
k
∈
N
0
{
ω
∈
Ω
:
lim sup
n
→
∞
|
ϕ
n
(
ω
)
−
p
|
>
1
k
}
)
=
0
{\displaystyle \operatorname {P} (\bigcup _{k\in \mathbb {N} _{0}}\{\omega \in \Omega :\limsup _{n\to \infty }|\phi _{n}(\omega )-p|>{\frac {1}{k}}\})=0}
Bis-subadditività
P
(
⋃
k
∈
N
0
{
ω
∈
Ω
:
lim sup
n
→
∞
|
ϕ
n
(
ω
)
−
p
|
>
1
k
}
)
{\displaystyle \operatorname {P} (\bigcup _{k\in \mathbb {N} _{0}}\{\omega \in \Omega :\limsup _{n\to \infty }|\phi _{n}(\omega )-p|>{\frac {1}{k}}\})}
≤
∑
k
∈
N
0
P
(
{
ω
∈
Ω
:
lim sup
n
→
∞
|
ϕ
n
(
ω
)
−
p
|
>
1
k
}
)
.
{\displaystyle \leq \sum _{k\in \mathbb {N} _{0}}\operatorname {P} (\{\omega \in \Omega :\limsup _{n\to \infty }|\phi _{n}(\omega )-p|>{\frac {1}{k}}\}).}
Mela, billi
P
{\displaystyle \operatorname {P} }
mhux negattiva, jekk nuru li
∀
k
∈
N
0
{\displaystyle \forall \ k\in \mathbb {N} _{0}}
P
(
{
ω
∈
Ω
:
lim sup
n
→
∞
|
ϕ
n
(
ω
)
−
p
|
>
1
k
}
)
=
0
{\displaystyle \operatorname {P} (\{\omega \in \Omega :\limsup _{n\to \infty }|\phi _{n}(\omega )-p|>{\frac {1}{k}}\})=0}
inkunu pprovajna l-liġi qawwija. L-ewwel ħa nipprovaw din għas-sottosuċċessjoni
ϕ
n
2
{\displaystyle \phi _{n^{2}}}
, jiġifieri
∀
k
∈
N
0
{\displaystyle \forall \ k\in \mathbb {N} _{0}}
P
(
{
ω
∈
Ω
:
lim sup
n
→
∞
|
ϕ
n
2
(
ω
)
−
p
|
>
1
k
}
)
=
0
{\displaystyle \operatorname {P} (\{\omega \in \Omega :\limsup _{n\to \infty }|\phi _{n^{2}}(\omega )-p|>{\frac {1}{k}}\})=0}
.
Biex nagħmlu dan, bil-lemma ta' Borel-Cantelli , biżżejjed li nivverifikaw li l-espressjoni li ġejja tikkonverġi
∑
n
=
1
∞
P
{
ω
∈
Ω
:
|
ϕ
n
2
(
ω
)
−
p
|
>
1
k
}
{\displaystyle \sum _{n=1}^{\infty }\operatorname {P} \{\omega \in \Omega :|\phi _{n^{2}}(\omega )-p|>{\frac {1}{k}}\}}
Bid-diżugwaljanza ta' Čebyšëv insibu li
∀
k
,
∀
n
{\displaystyle \forall \ k,\forall \ n}
P
(
{
ω
∈
Ω
:
|
ϕ
n
2
(
ω
)
−
p
|
>
1
k
}
)
≤
var
(
ϕ
n
2
)
k
2
=
k
2
p
(
1
−
p
)
n
2
{\displaystyle \operatorname {P} (\{\omega \in \Omega :|\phi _{n^{2}}(\omega )-p|>{\frac {1}{k}}\})\leq {\textrm {var}}(\phi _{n^{2}})k^{2}=k^{2}{\frac {p(1-p)}{n^{2}}}}
minn fejn:
∑
n
=
1
∞
P
(
{
ω
∈
Ω
:
|
ϕ
n
2
(
ω
)
−
p
|
>
1
k
}
)
≤
p
(
1
−
p
)
k
2
∑
n
=
1
∞
1
n
2
{\displaystyle \sum _{n=1}^{\infty }\operatorname {P} (\{\omega \in \Omega :|\phi _{n^{2}}(\omega )-p|>{\frac {1}{k}}\})\leq p(1-p)k^{2}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}}
Imma nafu li din is-serje tikkonverġi u għalhekk għandna li
∀
k
∈
N
0
{\displaystyle \forall \ k\in \mathbb {N} _{0}}
,
∀
k
∈
N
0
,
P
(
lim sup
n
→
∞
{
ω
∈
Ω
:
|
ϕ
n
2
(
ω
)
−
p
|
>
1
k
}
)
=
0.
{\displaystyle \forall \ k\in \mathbb {N} _{0},\operatorname {P} (\limsup _{n\to \infty }\{\omega \in \Omega :|\phi _{n^{2}}(\omega )-p|>{\frac {1}{k}}\})=0.}
Issa ninotaw li kull numru naturali n qiegħed bejn żewġ kwadrati konsekuttivi, jiġifieri,
∀
n
∈
N
,
∃
q
∈
N
{\displaystyle \forall \ n\in \mathbb {N} ,\exists \ q\in \mathbb {N} }
hekk li
q
2
≤
n
<
(
q
+
1
)
2
{\displaystyle q^{2}\leq n<(q+1)^{2}}
minn fejn inġibu
N
n
(
q
+
1
)
2
≤
ϕ
n
≤
N
n
q
2
.
{\displaystyle {\frac {N_{n}}{(q+1)^{2}}}\leq \phi _{n}\leq {\frac {N_{n}}{q^{2}}}.}
Issa ninotaw li
n
−
q
2
{\displaystyle n-q^{2}}
hi l-ikbar differenza possibbli bejn
N
q
2
{\displaystyle N_{q^{2}}}
u
N
n
{\displaystyle N_{n}}
, u għalhekk:
N
q
2
≤
N
n
≤
N
q
2
+
(
n
−
q
2
)
{\displaystyle N_{q^{2}}\leq N_{n}\leq N_{q^{2}}+(n-q^{2})}
u mela:
N
q
2
(
q
+
1
)
2
≤
N
n
(
q
+
1
)
2
≤
ϕ
n
≤
N
n
q
2
≤
N
q
2
+
(
n
−
q
2
)
q
2
.
{\displaystyle {\frac {N_{q^{2}}}{(q+1)^{2}}}\leq {\frac {N_{n}}{(q+1)^{2}}}\leq \phi _{n}\leq {\frac {N_{n}}{q^{2}}}\leq {\frac {N_{q^{2}}+(n-q^{2})}{q^{2}}}.}
Issa jekk nużaw
n
−
q
2
≤
(
q
+
1
)
2
−
q
2
{\displaystyle n-q^{2}\leq (q+1)^{2}-q^{2}}
, ikollna:
N
q
2
q
2
q
2
(
q
+
1
)
2
≤
ϕ
n
≤
N
q
2
q
2
+
(
q
+
1
)
2
−
q
2
q
2
.
{\displaystyle {\frac {N_{q^{2}}}{q^{2}}}{\frac {q^{2}}{(q+1)^{2}}}\leq \phi _{n}\leq {\frac {N_{q^{2}}}{q^{2}}}+{\frac {(q+1)^{2}-q^{2}}{q^{2}}}.}
Meta ngħaddu għall-limitu (
n
→
∞
⇒
q
→
∞
{\displaystyle n\to \infty \Rightarrow q\to \infty }
) u napplikaw ir-riżultat miksub għal
ϕ
n
2
{\displaystyle \phi _{n^{2}}}
, niksbu li, kważi ċertament:
p
=
p
lim
q
→
∞
q
2
(
q
+
1
)
2
≤
lim
n
→
∞
ϕ
n
≤
p
+
lim
q
→
∞
q
2
+
2
q
+
1
−
q
2
q
2
=
p
{\displaystyle p=p\lim _{q\to \infty }{\frac {q^{2}}{(q+1)^{2}}}\leq \lim _{n\to \infty }\phi _{n}\leq p+\lim _{q\to \infty }{\frac {q^{2}+2q+1-q^{2}}{q^{2}}}=p}
li ttemm il-prova.
^ Nistgħu ngħidulom ukoll varjabbli aleatorji jew varjabbli stokastiċi
^ Billi hemm ħafna verżjonijiet tat-transliterazzjoni mir-Russu ta' dan l-isem (Чебышёв): Chebychev, Chebyshov, Tchebycheff jew Tschebyscheff, qegħdin nużaw it-transliterazzjoni xjentifika (International Scholarly System).