p X - ambition

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FICHE 2 - corrigé ! Lois normales
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Exercice 1
"#!$%#&'()*+!,-!.-*'-/,+!-,0-1%'*+!
X
!&2'.-#1!,-!,%'!#%*3-,+!$+#1*0+!*0(2'1+!
N 0 ;1
( )
4!
5-#&!$+1!+6+*$'$+7!%#!-**%#('*-!,+&!*0&2,1-1&!-2!3',,')3+4!
!
9: 501+*3'#%#&!,+&!8*%/-/','10&!&2'.-#1+&!9!
!-:!!;!,<-'(+!(+!,-!$-,$2,-1*'$+7!%#!%/1'+#1!9!
p0≤X≤0,5
( )
≈0,191
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!/:!!"#!-!-,%*&!
p X ≤0,5
( )
=p X ≤0
( )
+p0≤X≤0,5
( )
≈0,5 +0,191 ≈0,691
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!$:!=-*!&>301*'+!(+!,-!$%2*/+!9!
p X >−0,5
( )
=p X <0,5
( )
=p X ≤0,5
( )
≈0,691
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!(:!!?-,$2,!(+!
p−1≤X≤0,5
( )
! !
!!!!!!!!"#!&-'1!@2+!
!!!!!!!
p−1≤X≤1
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=p µ − σ ≤X≤ µ + σ
( )
≈0,683
=2×p µ − σ ≤X≤0
( )
!
!!!!!!!5<%A!9!
p−1≤X≤0
( )
≈0,3415
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!!!!!!!
p−1≤X≤0,5
( )
=p−1≤X≤0
( )
+p0≤X≤0,5
( )
≈0,3415 +0,191 ≈0,533
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! +:!!
p X ≥1
( )
=p X ≤ −1
( )
=p X ≤0,5
( )
−p−1≤X≤ −0,5
( )
≈0,691−0,533 ≈0,158
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!B:!!?-,$2,!(+!
p X <−2
( )
!9!
!!!!!!"#!&-'1!@2+!9!
p X ≤ −2
( )
+p−2≤X≤2
( )
p µ −2 σ ≤X≤ µ +2 σ
( )
≈0,954
! "## $## +p X ≥2
( )
=1
!
!!!!!!C,%*&!9!
p X ≤ −2
( )
+p X ≥2
( )
≈1−0,954 ≈0,046
!"*!
p X ≤ −2
( )
=p X ≥2
( )
!D8-*!&>301*'+!(+!,-!$%2*/+E!
!!!!!"#!-!-,%*&!9!
2p X ≤ −2
( )
≈0,046
!+1!(%#$!
p X ≤ −2
( )
≈0,023
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!
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t
!9!
!-:!!
p X <t
( )
=0,8 ⇔t≈0,842
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!/:!!
p0≤X≤t
( )
=0,15 ⇔p X ≤0
( )
+p0≤X≤t
( )
=0,5 +0,15 =0,65 ⇔t≈0,385
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!$:!!
p X >t
( )
=0,9 ⇔1−p X ≤t
( )
=0,9 ⇔p X ≤t
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=0,1 ⇔t≈ −1,282
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!(:!!!
!!!!!!!
p−t≤X≤t
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=0,4 ⇔2p0≤X≤t
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=0,4
⇔p0≤X≤t
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=0,2 ⇔p X ≤0
( )
+p0≤X≤t
( )
=0,7
⇔p X ≤t
( )
=0,7 ⇔t≈0,524
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Exercice 2
Les$questions$suivantes$sont$indépendantes.$
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N 0 ;1
( )
!+1!
a
!2#!*0+,!1+,!@2+!!
p Y <a
( )
=0,7
4!
-:!!
p Y ≥ −a
( )
=p Y ≤a
( )
!!D8-*!&>301*'+!(+!,-!$%2*/+E!4!5<%A!!
p Y ≥ −a
( )
=0,7
! !
/:!!
p Y ≤ −a
( )
=1−p Y ≥ −a
( )
=0,3
!
!
$:!!!
!!!!!!
p Y <a
( )
=p Y ≤a
( )
=0,7 ⇔p Y ≤0
( )
+p0≤Y≤a
( )
=0,7
⇔0,5 +p0≤Y≤a
( )
=0,7 ⇔p0≤Y≤a
( )
=0,2
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Z
!&2'.-#1!,-!,%'!#%*3-,+!$+#1*0+!*0(2'1+!
N 0 ;1
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4!
501+*3'#+*!,+&!.-,+2*&!-88*%$F0+&!(+!
h
!+1!
t
!-2!$+#1')3+!8*)&!9!
-:!!
p Z ≤6t
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=0,99 ⇔6t≈2,33 ⇔t≈0,39
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/:!!"#!&-'1!@2+!
p−u0,05 <Z<u0,05
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=0,95 ⇔u0,05 ≈1,96
4!5<%A!!
2h≈1,96
!$-(!
h≈0,98
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Exercice 3
G#+!82$+!@2'!&+!1*%2.+!&2*!,<%*'H'#+!(<2#!-6+!H*-(20!+#!(0$'3)1*+!&+!8*08-*+!I!+BB+$12+*!2#!&-21!+#!,%#H2+2*!.+*&!2#!-21*+!
8%'#1!(+!,<-6+4!
"#!&288%&+!@2+!,<-/&$'&&+!(2!8%'#1!%A!*+1%3/+!,-!82$+!&2'1!,-!,%'!
N 0 ;1
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4!"#!-**%#('*-!,+&!*0&2,1-1&!-2!3',,')3+4!
9: ;!,<-'(+!(+!,-!$-,$2,-1*'$+7!%#!%/1'+#1!9!
p−1≤X<2
( )
≈0,819
4!
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p X =0
( )
=0
4!
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501+*3'#%#&!9!
p−1≤X
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+p X ≥1
( )
!9!
"#!&-'1!@2+!
p−1≤X≤1
( )
=p µ − σ ≤X≤ µ + σ
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≈0,683
4!5+!8,2&!9!
p−1≤X
( )
=p X ≥1
( )
!D&>301*'+!(+!,-!$%2*/+E!
N1!+#B'#!9!
p X ≤ −1
( )
+p−1≤X≤1
( )
+p X ≥1
( )
=1
!
%#!-!(%#$!9!
0,683+2p X ≥1
( )
≈1⇔2p X ≥1
( )
≈1−0,683 ⇔2p X ≥1
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≈0,317
4!
J-!82$+!-!OM7PQ!(+!$F-#$+!(+!8-*$%2*'*!8,2&!(+!M(34!
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"#!&-'1!@2+!
p−u0,01 ≤X≤u0,01
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≈0,99
!8%2*!
u0,01 =2,58
4!!
C'#&'!,+!$F-1!-!RRQ!(+!$F-#$+!(<0.'1+*!@2+!,-!82$+!#+!*+1%3/+!&2*!,2'7!+#!&+!8,-S-#1!I!T7UV!(3!(+!$+,,+W$'4!
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Exercice 4
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2#+!.-*'-/,+!-,0-1%'*+!&2'.-#1!,-!,%'!#%*3-,+!$+#1*0+!*0(2'1+!
N 0 ;1
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4!
"#!(%##+*-!,+&!*0&2,1-1&!-**%#('&!I!
10−4
!8*)&4!
!
9: J-!8*%/-/','10!(+!,<0.0#+3+#1!K!,+!$%381+!(+!X-6!+&1!I!(0$%2.+*1!L!+&1!!
p X < 0
( )
=p X ≤0
( )
=0,5
4!
!
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p0,2 ≤X≤0,5
( )
≈0,112
4!
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/:!J-!8*%/-/','10!(+!,<0.0#+3+#1!K!X-6!-!2#!(0$%2.+*1!$%38*'&!+#1*+!MZZ!+1!\ZZ!+2*%&!L!+&1!9!
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p−0,6 ≤X≤ −0,1
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≈0,186
4!
!!!!![,!>!-!(%#$!MV7\Q!(+!$F-#$+!@2+!X-6!-'1!2#!(0$%2.+*1!$%38*'&!+#1*+!MZZ!+1!\ZZ!+2*%&4!
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501+*3'#%#&!,-!8*%/-/','10!@2<',!&%'1!I!(0$%2.+*1!(+!8,2&!(+!UZZ!+2*%&!&-$F-#1!@2<',!-!*+S2!2#!]X]!9!
"#!$-,$2,+!(%#$!
pX≤0
( )
−0,5 ≤X
( )
!9!!
pX≤0
( )
−0,5 ≤X
( )
=p X ≤0
( )
et −0,5 ≤X
( )
( )
p X ≤0
( )
=p−0,5 ≤X≤0
( )
p X ≤0
( )
≈0,191
0,5
≈0,383
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