! " # $ $
%! ! % $ $ & $ $ '! (
"# % $ % $ ' ) *
+ ,
%. / ' '
10 $
$
$
%
$ $ 0 1
$ 2
P (
) =
1 1
1 1
2 2 2 2
× + × + × + ×
6 6 6 6 6 6 6 6
=
1+1+4+4
10
5
=
=
36
36
18
3 X $ μ = −4 4 ! * 5 X ! P (−10 < X < 2) ≥ 0.5
15
0 $ * 5 P (|X − μ| ε) σ2
.
ε2
0 P (−10 < X < 2) ≥ 0.5 0 ! −μ !
P (−10 − (−4) < X − μ < 2 − (−4)) 0.5 ⇒ P (−6 < X − μ < 6) 0.5
0 $ $ P (|X − μ| < 6) ≥ 0.5 +
P (|X − μ| < 6) = 1 − P (|X − μ| 6)
1 − P (|X − μ| 6) 0.5 ⇒ P (|X − μ| 6) 0.5.
& $ % ε = 6 * 5 $ P (|X − μ| 6) 0.5 =
σ2
⇒ σ 2 = V ar(X) = 36 × 0.5 = 18.
62
6
*
15 '! 78 $ $% ! 9 : 0 $% ! ! 2
!" # #$ % # $&
$ % 78 ! ! $ 1/60 $ 78 $ 1/60 μ = 47 ×
1 ∼
= 0.78.
60
$ P (
) = 1 − P ( ).
0
1
0.78 −0.78 ∼
e−0.78 +
e
) = 0.78
= 0.8159627001.
0!
1!
P (
) = 1 − 0.8159627001 ∼
= 0.1840372999.
15 $ !5 & 1! ) ;8 ! <
=
& %
$ > ! %=
$ $
1! 7 % %! 1! ?
$
& ! $ @A " !#2
!" # #$ ' # $&
0 $ "X # ! $! 1!
√
$ μ = 400 × 0.9 = 360 σ = 400 × 0.9 × 0.1 = 6
0 $ P (X 370) 0 370.5 P (
Z=
370.5 − 360
= 1.75.
6
+
! P (Z 1.75) = 0.5 + 0.4599 = 0.9599 (:
96%).
1.75
;
*
15
3 B
$
X Y ! f (x, y) =
1
(2x
4
0,
+ y), 0 x 1, 0 y 2,
$.
< "
# X ! Y
+ X ! Y μ∗1
= E[X|Y = y] =
∞
−∞
2
f2 (y) =
−∞
f (u, y)du =
1
1
(2u
0 4
0,
= 0.5
xf (x|y)dx
$ f (x|y) = ff(x,y)
f2(y) =
(y)
∞
∞
−∞
f (u, y)du
+
+ y)du = 14 (u2 + uy)10 = 14 (1 + y), 0 y 2,
$,
$ 0 y 2
f (x|y) =
f (x,y)
f2 (y)
=
0,
1
(2x+y)
4
1
(1+y)
4
=
2x+y
,
1+y
0 x 1,
x.
$ f2(y) = 0 +
f (x|0.5) =
4x + 1
2x + 0.5
=
.
1 + 0.5
3
$ X ! Y
= 0.5 1
∞
1 4x + 1
11
1 4 3 1 2
∗
μ1 = E[X|Y = 0.5] =
xf (x|0.5)dx =
x
= .
dx =
x + x
3
3 3
2
18
−∞
0
0
3 X $ μ c +
$ 10
E[(X − c)2 ] = V ar(X) + (c − μ)2 .
E[(X − c)2 ] = E[(X − c + μ − μ)2 ] = E[{(X − μ) − (c − μ)}2 ]
= E[(X − μ)2 + (c − μ)2 − 2(X − μ)(c − μ)]
= E[(X − μ)2 ] +(c − μ)2 − 2(c − μ) E[(X − μ)]
0
V ar(X)
2
= V ar(X) + (c − μ) .
7
*
20 *
$! 7 7 0
%! % $
" $
! ! # * X Y < Cov(X, Y ) V ar(X + Y )
Cov(X, Y ) = E(XY ) − E(X)E(Y ) 3 X
% 7 6 ; 7 $ 1/4 5
1
E(X) = (1 + 2 + 3 + 4) =
4
2
Y E(Y ) = 5/2 C (X, Y )
% $ 1/6 $! D (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)
*
! XY % $! 2, 3, 4, 6, 8, 12,
< 1
35
E(XY ) = (2 + 3 + 4 + 6 + 8 + 12) = .
6
6
Cov(X, Y ) =
5
35 5 5
− × =− .
6
2 2
12
V ar(X + Y ) $ V ar(X + Y ) = V ar(X) + V ar(Y ) − 2Cov(X, Y ).
+
1
V ar(X) = V ar(Y ) =
4
5
1−
2
2
2 2 2 5
5
5
5
+ 2−
+ 3−
+ 4−
=
2
2
2
4
35
5
5 5
= .
V ar(X + Y ) = + − 2 ×
4 4
6
3
*