[PDF] [PDF] 1 Université Mohammed Premier Échantillonnage et Estimation

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???????2200??????? ?????? ??97%? ???? ?? ???? ????? ????? ??? ??= 144? ???? ?? ???? ???? ?? ?? ??????50? f(x;) := 1 ex=; x0;

0;??????

P(X= 0) = C01000000:00400:996100000t0:

X400p398:4 N(0;1);

P(X >440) = 1P(X440)

1PX400p398:444040019:96

1FZ(2)

10:9772

X Tn102 pn N(0;1):

P(XT>2200) = 0:97

,P(Z >220010n2 pn ) = 0:97 ,P(Z <220010n2 pn ) = 0:03

220010n2

pn =z0:03

220010n2

pn =1:88 ,pn= 15:02

X==pn= 60=p50t8:485:

????? n= 5030? ?? ?????? ?? ??? ??? ?????? ??????? ???? Z=X

X=X1158:485tN(0;1):

P(X115) =P(X1158:4851151158:485) = 1P(Z0) =50%:

P(jX115j 5) =P(jZj 5=8:485)

=P(jZj 0:59) = 2FZ(0:59)1 =44:48%:

P(jX115j<5) =P(jZj<56

) =P(jZj 0:83) = 2FZ(0:83)1 =79:67%: ??????? ???? ?? ???? ???? ?? ?? ??????50??? ? (A)

X=rNnN1:2n

=r20005020001:144250 '20:1120:1120:11 (B)

X'20:2620:2620:26

(C)

X'20:3120:3120:31

P(jXj 25) =P(jXj=20:1125=20:11)

=P(jZj 1:24) = 2FZ(1:24)1'78:50%78:50%78:50%:

2FZ(1:2339)1'78:27%78:27%78:27%

p0:1p0:1(10:1)=100=p0:10:03 N(0;1):

P(p >0:05)P(p0:10:03>0:050:10:03)

P(Z >1:67)

= 1P(Z <1:67) = 1FZ(1:67) =FZ(1:67) =0:9525:

E(X) =1

Z +1 0 xex=dx = [xex=]+10+Z +1 0 = 0 + [ex=]+10 =(e1=e0=) =(01)

E(X) =)^=X:

L(X1;:::;Xn;) =nY

i=1f(Xi) 1 nexp(nX i=1X i=): L () = lnL(X1;:::;Xn;) =nlnnX i=1(Xi=): 8>< :@L (^) = 0; 2L@

2(^)<0

@L (^) = 0, n=^+nX i=1(Xi=^2) = 0,^=X: 2L@

2(^) =n=^22nX=^3=n=^2<0:

(n)=X:

E(^) =E(X) =E(X) =:

V(^) =V(X) =V(X) =V(X)n

=2n n7!+1!0:

V(X) =E(X2)[E(X)]2

1 Z +1 0 x2ex=dx2 = 222=2:

95%? ?? ???? ???? ???XN(;2)????

XN(;2=n);

P[z1=2X=

pn z1=2] = 1

P[1:96pn

X1:96pn

] = 0:95

P[X1:96pn

X+ 1:96pn ] = 0:95

P[X1:96pn

x+ 1:96pn ] = 0:95

I:C(0:95) = [x1:96pn

; x+ 1:96pn = [24:51:961:5=3;24:5 + 1:961:5=3] p= 88=120 = 0:73: ??????? n= 120>30? ?? ??? ????? ? ppq p(1p)n

N(0;1)

)P[z1=2ppq p(1p)n z1=2] = 1 )P[^prp(1p)n

2:58pp+rp(1p)n

2:58] = 0:99

I

0:99(p) = [prp(1p)n

2:58; p+rp(1p)n

2:58] = [0:732:580:04;0:73 + 2:580:04] = [0:63;0:84]: I a0:03rp(1p)n z1=20:03 nz21=2(0:03)20:1971 n111458: I

0:9() =

xz0:95spn = [62:43;73:57]; ???X N(;2)?? ? I

0:9() =

xt8;0:95spn = [53:12;82:88]; p= 40=50 = 0:8 ?? ? ?? ????? ???? ?? ?????? n= 5030? ????? I

0:9(p) ="

pz0:95rp(1p)n = [70:73%;89:27%]: ??? ??? ?? ???? ?? ??????? ??? f= 50=200>5%? r200502001rp(1p)n = 0:86810:0566 =0:0491: I

0:9(p)[pz0:950:0491][73%;87%]:

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