A[B =A\B
A\B =A[B
A \ B = (A [ B) \ B = A \ (A \ B),
P (A)
(A [ B) \ C = (A \ C) [ (B \ C).
0
P (⌦) = 1
P (A [ B) = P (A) + P (B)
P (A) = 1
A\B =;
P (A)
P (B \ A) = P (A [ B)
P (A) = P (B)
P (A [ B) = P (A) + P (B)
P (A|B) = P (A \ B)/P (B)
P (A \ B)
P (A \ B)
P (B) > 0
P (A \ B) = P (A|B)P (B)
A1 , A2 , . . . , Ak
P (B) =
n
X
⌦
P (B|Ai )P (Ai )
i=1
A
P (B) > 0
P (Aj |B) =
P (B|Aj )P (Aj )
P (B)
j = 1, . . . , k.
P (A \ B) = P (A)P (B)
B
A1 , A2 , . . . , Ak
P (A1 \ A2 \ · · · \ Ak ) = P (A1 )P (A2 ) . . . P (Ak ).
f (x) = P (X = x) pj = f (aj )
P (a  X  b) =
T = {x : P (X = x) > 0}
P
F (x) =
f (aj )
i
f (x)dx
a
T = {x : f (x) > 0}
Rx
F (x) =
f (t)dt
j:aj x
F (x) = P (X  x)
Rb
f (x, y) = fX (x)fY (y)
PP
E(g(X, Y )) =
g(ai , bj )f (ai , bj )
1
XY
P (a < X  b) = F (b) F (a)
P
R1
µ = E(X) =
aj f (aj )
µ = E(X) =
xf (x)dx
j
2
= V ar(X) = E((X
P
V ar(X) = (aj µ)2 f (aj )
j
E(g(X)) =
P
g(aj )f (aj )
E(g(X)) =
R1
X,Y /( X Y )
Cov(X + Y, U + V ) = Cov(X, U ) + Cov(X, V )
+Cov(Y, U ) + Cov(Y, V ),
g(x)f (x)dx
V ar(c1 X + c2 Y ) = c21 V ar(X) + c22 V ar(Y ) + 2c1 c2 Cov(X, Y )
1
j
E(a + bX) = a + bE(X)
µX µY
Cov(b + cX, a + dY ) = c d Cov(X, Y ),
µ)2 ) = E(X 2 ) µ2
R1
V ar(X) =
(x µ)2 f (x)dx
1
= Cov(X, Y ) = E((X µX )(Y µY )) = E(XY )
Cov(X, X) = V ar(X)
%X,Y =
1
j
V ar(a + bX) = b2 V ar(X)
E(c1 X1 + · · · + cn Xn ) = c1 E(X1 ) + · · · + cn E(Xn ).
V ar(c1 X1 + · · · + cn Xn ) = c21 V ar(X1 ) + · · · + c2n V ar(Xn )
X1 , . . . , Xn
E(X · Y ) = E(X)E(Y )
↵
F (q↵ ) = ↵
R
1
xn dx = n+1
xn+1 + C n 6=
X
f (x) = nx ⇡ x (1 ⇡)n x
x = 0, 1, . . . , n
Y
B(n, ⇡)
1
x
P o( )
f (x) = x! e
B(n, ⇡)
P o( )
E(X) = n⇡
V ar(X) = n⇡(1
E(X) =
V ar(X) =
x = 0, 1, 2, . . .
= n⇡
⇡)
n
30
⇡  0.05
f (x, y) = P (X = x, Y = y)
T = {(x, y) : P (X = x, Y = y) > 0}
P
fX (x) = P (X = x) =
f (x, bj )
j
fY |X (y|x) = f (x, y)/fX (x)
G(a, b)
f (x) = b 1 a I[a,b] (x)
F (x) = xb aa I[a,b] (x) + I(b,1) (x)
E(X) = (a + b)/2,
V ar(X) = (b
a)2 /12
Exp( )
N (µ,
2
2
)
>0
x
f (x) = e
F (x) = (1
I[0,1) (x) > 0
e x )I[0,1) (x)
h
i
2
f (x) = p2⇡1 2 exp (x2 µ)
2
F (x) = ((x µ)/ )
E(X) = 1/
V ar(X) = 1/ 2
E(X) = µ V ar(X) =
n
2
X̄n =
q↵ = µ + z ↵
1X
Xi
n i=1
2
Sn2 = SX
=
n
X
1
n
1 i=1
X̄n )2 =
(Xi
1
2
S̃n2 = S̃X
=
n
X
n i=1
X1 , . . . , X n
2
µ = E(Xi )
= V ar(Xi )
E(X̄n ) = µ
Zn
c| < ") ! 1
P (|Zn
X1 , . . . , X n
V ar(Xi ) < 1
X1 , . . . , Xn
E(Xi ) = µ
Zn =
z2R
n!1
Pn
c|
p
") ! 0
n ! 1.
1
n
n
X
1 i=1
(Xi
X̄n )(Yi
nµ
2
(z)
P (a  Zn  b) ⇡ (b)
X̄n
=p
2
2 (0, 1)
⇡)
n
T = g(X1 , . . . , Xn )
T
(Xi
X̄n ) =
n
⇣X
Xi2
n(X̄n )2
i=1
X i Yi
n
1
n
⌘
2
SX
nX̄n Ȳn
✓
E✓ (T ) ! ✓
E✓ (T ) ✓
✓
Tn
p
Tn ! ✓
✓
p
⌘
E✓ (T ) = ✓
n!1
an + bn Un + cn Vn ! a + b · u + c · v
✓
p
Tn ! ✓
h
p
µ
2 /n
h(Tn ) ! h(✓)
p
p
Un ! u Vn ! v an ! a bn ! b
a
Zn ⇠ N (0, 1)
(a)
n.
5 n 30 a, b 2 N0 0  a  b  n
⇣ b + 0.5 n⇡ ⌘
⇣ a 0.5 n⇡ ⌘
p
p
P (a  Hn  b) ⇡
,
n⇡(1 ⇡)
n⇡(1 ⇡)
⇣ b + 0.5 n⇡ ⌘
p
P (Hn  b) ⇡
.
n⇡(1 ⇡)
5 n(1
1
n
⇣X
2
1
q i=1
2 2
RX,Y = SX,Y / SX
SY
✓
µ = E✓ (Xi ) = g(✓)
Hn ⇠ B(n, ⇡) n⇡
Ȳn ) =
1
p
i=1 Xi
n
" > 0
X̄n ! E(Xi )
V ar(Xi ) =
p
/n.
Zn ! c
c
P (|Zn
2
V ar(X̄n ) =
SX,Y =
1
n
✓ ✓ = h(µ)
ˆ n = h(X̄n )
X̄n ⇥
cn ! c
µ
[ X̄n
µ
2
[X̄n
⇥
(n
h
µ
X̄n
(1 ↵)
p
p
2 /n, X̄ + z
2 /n ]
z1 ↵/2
n
1 ↵/2
n (z1 ↵/2 / )2
p
p
2
tn 1,1 ↵/2 Sn /n, X̄n + tn 1,1 ↵/2 Sn2 /n]
⇤
1)Sn2 / 2n 1,1 ↵/2 , (n 1)Sn2 / 2n 1,↵/2
T = pX̄n2 Ȳm 2 0
2
⇡
ˆ = X̄n
⇧
h
ˆ
⇧
2
(n
ˆ
ˆ + z1 ↵/2
⇧)/n,
⇧
ˆ = X/n
⇧
0.25(z1 ↵/2 / )2
n
⇤
ˆ
⇧(1
ˆ
⇧)/n
i
n
⇡
ˆ⇤ (1
⇡
ˆ⇤ )(z1 ↵/2 / )2
H1 : µ 6= µ0
2(1
µ0
H1 : µ < µ 0
(t)
H 0 : µ  µ0
H1 : µ > µ 0
t(n
X̄n µ0
T =p
2
N (0, 1)
Pn
i=1 xi Yi
P
n
2
i=1 xi
Ŷi = ˆ0 + ˆ1 xi ,
N (0, 1)
1
T = p X n⇡0
n⇡0 (1 ⇡0 )
T = (n
1)Sn2 / 02
T = pX̄n Ȳm 0 2
(1/n+1/m)Sp
1
2
Sp2 = n+m
(n · S̃X
+ m · S̃Y2 )
2
1
2
= n+m 2 ((n 1) · SX
+ (m 1) · SY2 )
2
(n
Yi =
ˆ2 =
n
n
X
2 i=1
Ni• ·N•j
n
2
1
(k
t(n
⌘
1+%X,Y
1 %X,Y
ln
1)(m
2)
N (0, 1)
30
(|t|))
(t)
1 · x i + "i
0+
i = 1, . . . , n
"i ⇠ N (0, 2 )
nx̄Ȳn
SX,Y
= 2 ,
nx̄2
sX
ˆ1 =
ˆ0 = Ȳn
ˆ1 · x̄.
Ŷ (x) = ˆ0 + ˆ1 · x,
(Yi
1)
t(n + m
Eij =
R
n
H0 : µ
X̄n µ0
T =p
2
Sn /n
p
(Nij Eij )2
Eij
j=1
H 0 : µ = µ0
N (0, 1)
1)
i=1
n, m
X̄n µ0
T =p
2
Sn /n
Pk Pm
2 · p X,Y2
⇣ 1 RX,Y
p
1+R
n 3
Z = 2 · ln 1 RX,Y
X,Y
⇤
⇡
q
=
T =
2
/n
2
2
⇤
30)
N (0, 1)
ˆ 1 = X/n ⇧
ˆ 2 = Y /m
⇧
⇤
2
↵)
q
ˆ
z1 ↵/2 ⇧(1
ˆ1 ⇧
ˆ2 0
⇧
ˆ 1 )/n+⇧
ˆ 2 (1 ⇧
ˆ 2 )/m
⇧1 (1 ⇧
T = pˆ
= LKI /2
(1 ↵)
µ
q
q
i
2
2
z1 ↵/2 S̃n /n, X̄n + z1 ↵/2 S̃n /n
(1
N (0, 1)
S̃X /n+S̃Y /m
SQR
n
Ŷi )2 =
=
n 2
n
1 2
S (1
2 Y
2
RXY
)=
n
n
1⇣ 2
S
2 Y
2
⌘
SX,Y
.
s2X
SQT = SQE + SQR
2)
SQT =
n
X
(Yi
Ȳn )2 ,
i=1
SQE =
n
X
(Ŷi
Ȳn )2 ,
SQR =
i=1
R2 =
SQE
=1
SQT
n
X
i=1
SQR
2
= RX,Y
.
SQT
(Yi
Ŷi )2 .
1)
ˆ0
ˆ 2ˆ0 =
n(
ˆ2
Pn
Pn
2
i=1 xi
,
2
nx̄2 )
i=1 xi
0
T0 =
ˆ0
0,0
T1 =
ˆ ˆ0
0 =
(z)
ˆ2
2
i=1 xi
ˆ 2ˆ1 = Pn
1
H0 :
[ ˆj
ˆ1
nx̄2
(1.75) = (1.7 + 0.05) ⇡ 0.9599
.
z↵ =
ˆ1
1,0
ˆ ˆ1
0,0
H0 :
0
1
.
1 =
tn 2,1 ↵/2 · ˆ ˆj , ˆj + tn 2,1 ↵/2 · ˆ ˆj ],
1,0
t(n
2)
j = 0, 1.
Ŷ (x) = ˆ0 + ˆ1 · x Ỹ (x) = ˆ0 + ˆ1 · x + "x
ˆŶ2 (x) = ˆ 2
[Ŷ (x)
[Ŷ (x)
⇣1
n
(x
+ Pn
(1
ˆỸ2 (x) = ˆŶ2 (x) + ˆ 2 .
1
↵
tn 2,1 ↵/2 · ˆỸ (x) , Ŷ (x) + tn 2,1 ↵/2 · ˆỸ (x) ].
0+
1 xi,1 + · · · +
p xi,p + "i .
j
Tj = ( ˆj
2
x̄)2 ⌘
,
x̄)2
tn 2,1 ↵/2 · ˆŶ (x) , Ŷ (x) + tn 2,1 ↵/2 · ˆŶ (x) ],
Yi =
N (0,
i=1 (xi
j,0 )/ˆ ˆj
j
=
j,0
t(n
p
1)
)
↵)
[ ˆj
tn p 1,1 ↵/2 · ˆ ˆj , ˆj + tn p 1,1 ↵/2 · ˆ ˆj ],
j = 0, . . . , p.
"i ⇠
z1 ↵
↵ 2 (0, 0.5)
z
0
( z) = 1
(z)
z0.975 ⇡ 1.96
2
t
t20,0.99 ⇡ 2.528
n > 40
tn,↵ ⇡ z↵
2
n
2
10,0.95 ⇡ 18.307
n