37. FX (x)
fX (x)
X1, X2, · · · , Xn FX (x) fX (x)
X(1), X(2), · · · , X(n) Xi
X(j), j = 1, 2, · · · , n
fX(j)
=
n!
(j 1)!(n j)!
fX(x)[FX(x)]j 1
[1 FX(x)]n j
1 FX(x)FX(x)
x
fX (x)
i
j
38. Y = #{Xj, j = 1, 2, · · · , n|Xj x}
x
Y個
Zj =
(
1 if {Xj x}
0 otherwise
x
Z4=1
Z3=1
Z9=1
Z8=1 Z6=1
Z2=1 Z1=0
Z5=0
Z7=0
Y =
nX
j=1
Zj
39. P(Zj = 1) = Pi = FX(x)
1
1O
FX(x)
Xi
Zi
Zi
xP(Zj = 1) = Pi = FX (x)
x
Zi
Pi
40. FX(j)
(x) = P(Y j) =
nX
k=j
✓
n
k
◆
[FX (x)]k
[1 FX(x)]n k
1
FX (x) = P6
x
Y j
Xi x
41. FX(j)
(x) fX(j)
(x)
fX(j)
(x) =
dFX(j)
(x)
dx
(f · g)0
= f0
g + fg0
=
d
dx
nX
k=j
✓
n
k
◆
[FX(x)]k
[1 FX(x)]n k
[f(g(x))]0
= f0
(g(x))g0
(x)
=
nX
k=j
✓
n
k
◆
kfX(x)[FX(x)]k 1
[1 FX(x)]n k
(n k)fX(x)[FX(x)]k
[1 FX(x)]n k 1
=
✓
n
k
◆
jfX (x)[FX (x)]j 1
[1 FX (x)]n j
+
nX
k=j+1
✓
n
k
◆
kfX (x)[FX (x)]k 1
[1 FX (x)]n k
n 1X
k=j
(n k)fX (x)[FX (x)]k
[1 FX (x)]n k 1
42. =
n!
(j 1)!(n j)!
fX (x)[FX (x)]j 1
[1 FX (x)]n j
+
n 1X
k=j
✓
n
k + 1
◆
(k + 1)fX (x)[FX (x)]k
[1 FX (x)]n k 1
n 1X
k=j
✓
n
k
◆
(n k)fX (x)[FX (x)]k
[1 FX (x)]n k 1
=
✓
n
k
◆
jfX (x)[FX (x)]j 1
[1 FX (x)]n j
+
nX
k=j+1
✓
n
k
◆
kfX (x)[FX(x)]k 1
[1 FX (x)]n k
n 1X
k=j
✓
n
k
◆
(n k)fX(x)[FX (x)]k
[1 FX (x)]n k 1
43. =
n!
(j 1)!(n j)!
fX (x)[FX(x)]j 1
[1 FX (x)]n j
+
n 1X
k=j
✓
n
k + 1
◆
(k + 1)fX(x)[FX (x)]k
[1 FX (x)]n k 1
n 1X
k=j
✓
n
k
◆
(n k)fX (x)[FX (x)]k
[1 FX(x)]n k 1
=
n!
(j 1)!(n j)!
fX(x)[FX(x)]j 1
[1 FX(x)]n j
+ fX(x)[FX(x)]k
[1 FX(x)]n k 1
0
@
n 1X
k=j
✓
n
k + 1
◆
(k + 1)
n 1X
k=j
✓
n
k
◆
(n k)
1
A
= 0
✓
n
k + 1
◆
(k + 1) =
n!
k!(n k 1)!
=
✓
n
k
◆
(n k)
=
n!
(j 1)!(n j)!
fX(x)[FX(x)]j 1
[1 FX(x)]n j
44. 【証明】(cont.)
X1, X2, · · · , Xn
fX(j)
(x) =
n!
(j 1)!(n j)!
fX (x)[FX (x)]j 1
[1 FX (x)]n j
fX(x) =
(
1 0 < x < 1
0 otherwise
FX (x) =
8
><
>:
0 x 0,
x 0 < x < 1,
1 x 1
= 1, (0 < x < 1) = x, (0 < x < 1)
fX(j)
(x) =
(
0 otherwise
n!
(j 1)!(n j)!
xj 1
(1 x)n j
, 0 < x < 1
n = j + i 1 ! n j = i 1
!
n!
(j 1)!(i 1)!
xj 1
(1 x)i 1