∑ ∑
!
Formulaire*de*probabilités*
!
!
1. Formules*de*calcul*des*probabilités*
*
"#$%&'()!*(!+,)(*
)(1)(
)()()()(
)(1)(
CC
C
BAPBAP
BAPBPAPBAP
APAP
∩−=∪
∩−+=∪
−=
!
!
"#$%&'()!*(!-,'-&'!-#%+./,0#.$(!
1 1! 0! 1 ... . 2)(n . 1)-(n .n n!
!)!-(
!
=
)!(
!
! ==−=
−
== mmn
n
C
mn
n
AnP m
n
m
nn
!
!
1$#+,+.'.02!-#/*.0.#//(''(!3
)(
)(
)( BP
BAP
BAP ∩
=
!
1$#+,+.'.02)!-#%4#)2()!
)()()()()( APABPBPBAPBAP ==∩
!
5/*24(/*,/-(!3
A et B sont indépendants ssi P(A!B)=P(A).P(B)
!
!
"#$%&'(!*(!6,7()!3!
P B A
P A B P B
P A
( ) ( ) ( )
( )
=
!
!
1$#+,+.'.02)!0#0,'()!3!
)()( )()()( C
CBPBAPBPBAPAP +=
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!80!).!69:!6;<<<!6=!>#$%(/0!&/!)7)0?%(!-#/0$,*.-0#.$(!*@2A2/(%(/0)3!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
)()()()(
11 ∑∑ ==
=∩=
k
i
ii
k
i
iBPBAPBAPAP
! !
!
! !
2. Variables*aléatoires*:*généralités*
*
B#7(//(!#&!()42$,/-(!%,0C2%,0.D&(!*@&/(!A,$.,+'(!*.)-$?0(!3!!
∑
=
===
k
i
ii xXPxXE
1
)()(
µ
!
!
E,$.,/-(!(0!*2A.,0.#/!)0,/*,$*!*@&/(!A,$.,+'(!*.)-$?0(!3!!
∑
=
=−=−==
k
i
ii xXPxXEXV
1
222 V(X)=et )()())(()(
σµµσ
!
!
F#(>>.-.(/0!*(!-#$$2',0.#/!ρ!!
11où
Y)cov(X,
)))((( ≤≤−=
−−
=
ρ
σσσσ
µµ
ρ
YXYX
YX YXE
!
!
!
8)42$,/-(!(0!A,$.,/-(!*(!>#/-0.#/)!'./2,.$()!*(!A,$.,+'()!,'2,0#.$()!3!
E a bX abE X V a bX b V X
EaX bY aE XbE Y
( ) ( ) ( ) ( )
( ) ( ) ( )
+ = + + =
+ = +
et 2
!
!!!!!!!!!!!!! !
VaX bY a V X b V Y( ) ( ) ( )+ = +
2 2
).!G!(0!H!)#/0!*(&I!A<,<!./*24(/*,/0()!
!
3. Variables*aléatoires*:*lois*classiques*
*
*
Nom*
Valeurs*possibles*
ou*Domaine*
P(X=x)*
ou*densité*
Moyenne*
E(X)*
Variance*
V(X)*
J/.>#$%(!*.)-$?0(!
G!~*JKL,:=M!
I!N!,:!,O9:!<<<:,OL=P9M!
9Q=!
,OL=P9MQ;!
L=;P9MQ9;!
6($/#&''.!!
G!~!!6(LπM!
I!N!R:!9!
xx −
−1
)1(
ππ
!
π!
π!L9PπM!
6./#%.,'(!!
G!~!!6.L/: πM!
I!N!R:!9:!;:!<<<<<:/!
)(
)1( xnxx
n
C−
−
ππ
!
/π!
/π(9PπM!
J/.>#$%(!F#/0./&(!
G!~*JL,:+M!
K#%,./(!N![,:+]!
f(x)%=%1/(b*a)!
L,O+MQ;!
L+P,MSQ9;!
!
T#$%,'(!U2*&.0(!
V!W!TLR:9M!
!
K#%,./(!N!U!N]PX:OX[!
2
2
2
1
)(
z
ezf −
=
π
!
!
R!
!!
9!
!
T#$%,'(!!
G!~*TLµ:σSM!
!
K#%,./(!N!U!N]PX:OX[!
2
2
2
)(
2
1
)(
σ
µ
σπ
−
−
=
x
exf
!
!
µ!
!!
σS!
*
4. Calcul*de*probabilité*sur*une*variable*aléatoire*Normale*
*
Y#.(/0!G!&/(!A,$.,+'(!,'2,0#.$(!T#$%,'(!D&('-#/D&(!TLµ:σ;M!(0!V!',!A,$.,+'(!,'2,0#.$(!/#$%,'(!$2*&.0(!TLR:9M!
!
5/0($A,''(!*,/)!'(D&('!#/!0$#&A(!Z[\!*(!',!4#4&',0.#/!'.2(!]!G!3!!
[ ]
σµσµ
96.1,96.1 +−
!
Y0,/*,$*.),0.#/!!!
σ
µ
−
=X
Z
!!!!^$,/)>#$%,0.#/!./A($)(!]!',!)0,/*,$*.),0.#/!
σµ
ZX +=
!!
F,'-&'!*@&/(!4$#+,+.'.02!
!
"
#
$
%
&−
≤=≤
σ
µ
x
ZPxXP )(
!
F,'-&'!*&!4($-(/0.'(!I4!0('!D&(!
pxXP p=≤)(
!3!
σµ
pp zx +=
!
!
5. Combinaison*linéaire*de*variables*aléatoires*Normales*
*
Y#.(/0!G9!(0!G;!*(&I!A,$.,+'()!,'2,0#.$()!./*24(/*,/0()!TLµ9:σ9
;M!(0!TLµ;:σ;
;M:!!!
! !
abX cX N a b c b c+ + + + +
1 2 1 2
2
1
2 2
2
2
est ( , )
µ µ σ σ
!
Y#.(/0!G9:!G;<<<<<!G/!/!A,$.,+'()!,'2,0#.$()!TLµ:σ;M!./*24(/*,/0()!
! !
Sn=Xi
i=1
n
!
()0!
),( 2
σµ
nnN
!(0!!!!!!!!!!
∑
=
=
n
i
i
X
n
X
1
1
()0!
),(
2
n
N
σ
µ
!
!
_< Somme*et*moyenne*de*variables*aléatoires*iid*L*Théorème*central*limite!
*
Y#.(/0!G9:!G;:!<<<!G/!/!A,$.,+'()!,'2,0#.$()!./*24(/*,/0()!(0!.*(/0.D&(%(/0!*.)0$.+&2()!L..*M!*(!!
%#7(//(!µ!(0!*(!A,$.,/-(!σ;!
!
Sn=Xi
i=1
n
!
()0!,44$#I.%,0.A(%(/0!&/(!
),( 2
σµ
nnN
!D&,/*!/!()0!`$,/*!! !
∑
=
=
n
i
i
X
n
X
1
1
!!()0!,44$#I.%,0.A(%(/0!&/(!
),(
2
n
N
σ
µ
D&,/*!/!()0!`$,/*<!
!
* *
!
a< Théorème*central*limite*appliqué*à*la*variable*aléatoire*Binomiale!
*
J/(!A<,<!G!*(!*.)0$.+&0.#/!+./#%.,'(!6.L/:πM!()0!,44$#I.%,0.A(%(/0!&/(!N(n
π
,n
π
(1*
π
))!).!/!()0!`$,/*!L/π!b
