TD 4. La loi normale

XN(m, σ)
Z=1
σ(X−m)N(0,1)
a<b
P[a≤Z≤b] = Pha≤1
σ(X−m)≤bi
=Phm+σa ≤X≤m+σbi
=Zm+σb
m+σa
1
σ√2πexp −(x−m)2
2σ2dx
u=(x−m)
σ
P[a≤Z≤b] = Zb
a
1
√2πexp −u2
2du
ZN(0,1)
XN(0,1) P[X≤1.62] P[X≥
−0.52] P[−1< X < 1] u P [X≤u] = 0.334
P[X≤1.62] = 0.9474
X−X
P[X≥ −0.52] = P[−X≤0.52] = P[X≤0.52] = 0.6985
P[−1< X < 1] = P[X < 1] −P[X < −1] = P[X < 1] −P[−X > 1]
=P[X < 1] −P[X > 1] = 2.P [X < 1] −1=0.6826
u=−0.4289
m= 2 σ= 0,012
2,000 ±0,012
p= 1 −P[2 −0,012 ≤X≤2+0,012]
= 1 −Ph−1≤X−2
0,012 ≤1i
= 1 −P[−1≤Y≤1] YN(0,1)
= 2(1 −P[Y≤1]) = 0,3174
X
P[X≤3] = 0,5517 et P[X≥7] = 0,0166
X

XN(m, σ)
0,5517 = P[X≤3] = PhX−m
σ≤3−m σi=PhY≤3−m σi
UN(0,1) P[Y≤0,13] = 0,5517
(3 −m)/σ = 0,13
0,0166 = P[X≥7] = PhY≥7−m
σi= 1 −PhY≤7−m σi
PhY≤7−m
σi= 0,9834
(7 −m)/σ = 2,13 σ= 2 m= 2,74
m
0,90 = P[X≥250] = PhX−m
3≥250 −m 3i=PhY≥250 −m 3i
YN(0,1) P[Y≤
−1,2816] = 0,10 m= 253,84
TDTST B
RDRS
P[RD] = P[TD≥15] = P[TD−13
3≥15−13 3]=1−P[Y≤0,67] = 1 −0,7486 = 0,2514
P[RS] = P[(T≥18) (T < 18 B≥10)] = P[T≥18] + P[T < 18]P[B≥10]
P[RS]=0,2922
P[RD∪RS] = 1 −P[RD∩RS]=1−0,2514 ×0,2922 = 0,929

P[X > 100] P[100 ≤X≤150]
p= 0,56 n= 200
X
B(200; 0,56) k= 0,1, ..., 200 P[X=k] = 200
k0,56k0,44200−k
n= 200 ≥20 np ≥5n(1 −p)≥5X
N(112,7,02)
P[X > 100] = 1 −P[X≤100] = 1 −P[Y≤ −1,71] = P[Y≤1,71] = 0,9564
P[100 ≤X≤150] = P[X≤150] −P[X≤99] = P[Y≤5,41] −P[Y≤ −1,85] = 1 −P[Y≤ −1,85] =
P[Y≤1,85] = 0,9678
n X
N(n.0,56; √n.0,56.0,44) 0,95 = P[X≥100] = P[Y≥100−n.0,56
√n.0,56.0,44 ]
100−n.0,56
√n.0,56.0,44 = 1,6449 n= 200
n
A Acp1−p
K A
K
F=K/n A
F
F n
F
7% 23%
F15% 15%
A
n p
KB(n;p)
K=Pn
i=1 YiYip
E[F] =
n
X
i=1
E[Yi]/n =p, Var(F) =
n
X
i=1
Var(Yi)/n2=p(1 −p)/n
F p F
N(p, qp(1−p)
n)

n= 400 p= 0,2
P[0,07 ≤F≤0,23] = Ph0,07 −0,2
p0,2×0,8/400 ≤F−0,2 p0,2×0,8/400 ≤0,23 −0,2 p0,2×0,8/400i
=P[−6,5≤Y≤1,5] YN(0,1)
=P[Y≤1,5] = 0,9332
P[F≥0,2] = 0,5
P[F≤0,15] = P[Y≤ −2,5] = 1 −P[Y≤2,5] = 0,0062
XE(1) P[X≤2] P[X > 0.5]
Y= 3X
X f(x) = exp(−x)R+0R−
P[X≤2] = Z2
0
exp(−x)dx= 1 −exp(−2) = 0.8647
P[X > 0.5] = Z+∞
0.5
exp(−x)dx= exp(−0.5) = 0.6065
a > 0y= 3x
P[Y≤a] = P[3X≤a] = P[X≤a/3] = Za/3
0
exp(−x)dx=Za
0
1
3exp(−y/3)dy
Yexp(−y/3)/3E(1/3)