(&!, F, P)
X : &! Rn
Rn
{É " &! : X(É) " B}
B
X : &! Rn
{É " &! : X(É) " B} " F
B
 X : &! Rn
Rn
R
R2
X(É) É
É
FX {É " &! : X(É) " B} B
FX
X
FX ‚" F
(&!, F, P) X
à F X
FX FX Ã
X : &! Rn n > 1
X(É) = [X1(É), X2(É), . . . , Xn(É)]
Xi : &! R i = 1, 2, . . . , n
X n
X1 X2 Xn X : &! Rn
X(É) = [X1(É, X2(É), . . . , Xn(É)].
É"&!
n
X : &! Rn
(&!, F, P) B
{É : X(É) " B}
F
X B
PX
PX(B) = P({É : X(É) " B})
Rn
PX(Rn) = P({É : X(É) " Rn}) = P(&!) = 1.
B1
B2 . . . Bn Rn
n n n
PX Bi = P {É : X(É) " Bi} = P {É : X(É) " Bi}
i=1 i=1 i=1
n n
= P({É : X(É) " Bi}) = PX(Bi).
i=1 i=1
PX
PX(B) = P({É : X(É) " B})
X
P
X P X <" P
(Rn, Bn, PX) Bn
Rn PX X
(&!, F, P)
X
(&!, F, P) Rn
(&!, F, P) (Rn, Bn, PX)
X
m m < n
n X
X
n
(-", x) x " R
(-", x) x = (x1, x2, . . . , xn) n
(-", xi) i = 1, 2, . . . , n
P X
(&!, F, P)
F : Rn < 0, 1 >
F (x) = P({É " &! : X(É) " (-", x)})
X
FX FX
X
FX(x) = FX ,X2,...,Xn(x1, x2, . . . , xn)
1
= P({É " &! : X1(É) " (-", x1), X2(É) " (-", x2), . . . , Xn(É) " (-", xn)})
= P(X1 < x1, X2 < x2, . . . , Xn < xn).
 FX : R < 0, 1 >
FX(x) = P(X < x),
FX,Y : R2 < 0, 1 >
FX,Y (x, y) = P(X < x, Y < y).
n
n
lim F (x1, . . . , xk-1, xk + h, xk+1, . . . , xn)
h0-
k"{1,2,...,n}
=F (x1, . . . , xk-1, xk, xk+1, . . . , xn)
k " {1, 2, . . . , n} (x1, x2, . . . , xn) " Rn
F (x1, . . . , xk-1, xk+h, xk+1, . . . , xn) e" F (x1, . . . xk-1, xk, xk+1, . . . , xn)
he"0
-" lim F (x) = 0
+" lim F (x) = 1
lim F (x1, . . . , xk-1, xk, xk+1, . . . , xn) = 0,
xk-"
k"{1,2,...,n}
lim F (x1, x2, . . . , xn-1, xn) = 1.
(x1,...,xn)(",...,")
F
F : Rn < 0, 1 >
m
 n X
FX B
PX(B) = P(X " B)
a b
P(X < a) = FX(a)
P(X e" a) = 1 - FX(a)
P(a d" X < b) = FX(b) - FX(a)
P(X = a) = lim FX(x) - FX(a)
xa+
P(X d" a) = lim FX(x)
xa+
P(X > a) = 1 - lim FX(x)
xa+
P(a d" X d" b) = lim FX(x) - FX(a)
xb+
P(a < X < b) = FX(b) - lim FX(x)
xa+
P(a < X d" b) = lim FX(x) - lim FX(x)
xb+ xa+
FX
{X e" a} = {X < a}
P(X e" a) = 1 - P(X < a) = 1 - FX(a).
a < b {É : X(É) < a}
{É : X(É) < b}
P(a d" X < b) = P((X < b) - (X < a))
= P(X < b) - P(X < a) = FX(b) - FX(a).
 (xn)
{a d" X < a+xn}
"
P(X = a) = P( {a d" X < a + xn}) = lim P(a d" X < a + xn)
n"
n
= lim (FX(a + xn) - FX(a)) = lim FX(x) - FX(a).
n"
xa+
{X d" a} = {X = a} *" {X < a} '" {X = a} )" {X < a} = ".
{X > a} = {X e" a} - {X = a} '" {X = a} ‚" {X e" a}.
P(X > a) = P(X e" a) - P(X = a)
= 1 - FX(a) - ( lim FX(x) - FX(a)) = 1 - lim FX(x).
xa+ xa+
Xi : &! R i = 1, 2, . . . , n
(&!, F, P) FX = FX ,X2,...,Xn
1
FX
i
X1, X2, . . . , Xn
x1, x2, . . . , xn " R
{X1 < x1}, {X2 < x2}, . . . {Xn < xn}
X1, X2, . . . , Xn
X1, X2, . . . , Xn
 x1, x2, . . . , xn " R
FX(x1, x2, . . . , xn) = FX (x1)FX (x2) . . . FX (xn),
1 2 n
B1, B2, . . . , Bn
{X1 " B1}, {X1 " B1}, . . . {Xn " Bn}
FX(x1, x2, . . . , xn) = P(X1 < x1, X2 < x2, . . . , Xn < xn)
= P({X1 < x1} )" {X2 < x2} )" · · · )" {Xn < xn})
= P(X1 < x1)P(X2 < x2) . . . P(Xn < xn)
= FX (x1)FX (x2) . . . FX (xn).
1 2 n
(-", x)
X1, X2, . . . , Xn
x1, x2, . . . , xn " R
P(X1 = x1, X2 = x2, . . . , Xn = xn) = P(X1 = x1)P(X2 = x2) . . . P(Xn = xn)
i
Xi : &! Rn i = 1, 2, . . . , m
(&!, F, P)
1 2
X1, X2, . . . , Xm x1 " Rn x2 " Rn
m
xm " Rn
{X1 < x1}, {X2 < x2}, . . . {Xm < xm}.
X
(&!, F, P) PX
 x " R
X
PX({x}) = P({É : X(É) = x}) = p > 0.
p x
X
SX
x0 " R X
F
x0
P(X = x0) = lim F (x) - F (x0).
xx+
0
X
PX(SX) = P(X " SX) = 1.
X
SX
X
SX = {x1} *" {x2} *" {x3} *" . . . .
PX(SX) = PX({x1}) + PX({x2}) + PX({x3}) + . . . .
X
PX({x1}) + PX({x2}) + PX({x3}) + · · · = PX(SX) = 1,
1
PX(SX) = PX({x1}) + PX({x2}) + PX({x3}) + · · · = 1,
X
 X
xi " SX pi
P(X = xi) = pi, xi " SX; i = 1, 2, . . .
X x1 x2 . . . xn
P(X = xi) p1 p2 . . . pn
X
B
PX(B) = pi,
i
i xi " B
B SX
X B
B = (B )" SX) *" (B - SX).
PX(B) = PX(B )" SX) + PX(B - SX).
(B )" SX) ‚" SX
B )" SX = {xj } *" {xj } *" {xj } *" . . . ,
1 2 3
xj " SX i = 1, 2, . . .
i
PX(B )" SX) = P(X = xj ) + P(X = xj ) + P(X = xj ) + · · · = pi.
1 2 3
i
PX(R - SX) = 0.
PX(R - SX) > 0
PX(R) = PX(SX) + PX(R - SX) = 1 + PX(R - SX) > 1.
0 d" PX(B - SX) d" PX(R - SX) = 0,
 (B - SX) ‚" (R - SX) PX(B - SX) = 0
PX(B) = PX(B )" SX) + PX(B - SX) = PX(B) = pi.
i
X
P(X = xi) = pi, x"SX, i = 1, 2, . . . ,
F (x) = pi,
i
i xi < x
B = (-", x)
B R
g
R
{x " R : g(x) " B} " B
B"B
X g
Y = g(X)
A = {x " R : g(x) " B}
B {É " &! :
X(É) " A}
{É " &! : Y (É) " B} = {É " &! : g(X(É)) " B} = {É " &! : X(É) " A} " F.
Y
Y X
 SX
X
P(X = xi) = pi, xi " SX
g SX SY Y = g(X)
P(Y = yi) = pk, yi " SY ,
k
k g(xk) = yi
yi " SY xk , xk , . . .
1 2
g(xk) = yi SX Y = yi
{Y = yi} = {X = xk } *" {X = xk } *" . . . .
1 2
P(Y = yi) = P(X = xk ) + P(X = xk ) + · · · = pk.
1 2
k
r r = 1, 2, . . . c
X P(X = xi) = pi i = 1, 2, . . .
"
µr(c) = E(X - c)r := (xi - c)rpi,
i=1
X
 c = 0
"
mr = EXr := xrpi,
i
i=1
c = m1 c = EX
"
µr = E(X - EX)r := (xi - EX)rpi.
i=1
X E(X) EX
g(X) g
X
P(X = xi) = pi i = 1, 2, . . .
"
Eg(X) = g(xi)pi,
i=1
X
P(X = xi) = pi i = 1, 2, . . . f g
a " R k " N
Ef(X) Eg(X)
E(f(X) + g(X)) = Ef(X) + Eg(X).
f(X)
g(X)
" " "
| f(xi) + g(xi) | pi d" | f(xi) | pi + | g(xi) | pi.
i=1 i=1 i=1
E(a) = a.
P(X = a) = 1
E(a) = aP(X = a) = a.
X
a " R k " N
E(aX)k = akEXk.
k " N
" "
E(aX)k = (axi)kpi = ak xkpi = akEXk.
i
i=1 i=1
" "
EX (axi)kpi = ak xkpi
i
i=1 i=1
E(X) a, b " R
E(aX + b) = aEX + b.
E(X) E(X - EX) = 0
E(X - EX) = E(X) - E(EX) = 0.
Y = X - EX
X Y
"
2
|E(XY )| d" EX2EY .
Z = (X - aY )2 a " R
X Y
Z
2
EZ = EX2 - 2aE(XY ) + a2EY .
Z
EZ e" 0
2
a2EY - 2aE(XY ) + EX2 e" 0
 a
2
4 (E(XY ))2 - 4EX2EY d" 0.
EX EY
a, b " R E(aX + bY )
E(aX + bY ) = aEX + bEY.
X Y
E(XY )
E(XY ) = EX · EY.
X D2(X) D2X
D(X) DX
c = EX

D2X < E(X - c)2.
E(X - c)2 = E(X - EX + EX - c)2
= E(X - EX)2 + 2E(X - EX)(EX - c) + (EX - c)2
= D2X + (EX - c)2 > D2X.
 E(X - c)2
X c
D2X = EX2 - (EX)2.
D2X = E(X - EX)2
= E(X2 - 2XEX + (EX)2)
= EX2 - 2EXEX + (EX)2
= EX2 - (EX)2.
b
D2(X + b) = D2X.
D2(X + b) = E(X + b)2 - (E(X + b))2
= E(X2 + 2bX + b2) - (EX + b)2
= EX2 + 2bEX + b2 - (EX)2 - 2bEX - b2
= EX2 - (EX)2 = D2X.
b
D2(bX) = b2D2X.
D2(bX) = E(bX)2 - (E(bX))2
= b2EX2 - b2(EX)2 = b2D2X.
b " R
D2(b) = Eb2 - (Eb)2 = b2 - b2 = 0,
X
1
à Y = X
Ã
 X Ã
1 1 1
D2 X = D2X = Ã2 = 1.
à Ã2 Ã2
X µ
X-µ
à Y =
Ã
X - µ 1
E = (EX - µ) = 0.
à Ã
X - µ 1
D2 = D2X = 1.
à Ã2
xp
P(X d" xp) e" p, P(X e" xp) e" 1 - p; 0 < p < 1
p X
1
2
X Me X
1 1
P(X d" Me X) e" P(X e" Me X) e" .
2 2
X
Mo X
 p
P(X = x1) = p P(X = x2) = q; 0 < p < 1, p + q = 1.
x1 = 0 x2 = 1
p
"
1 1
P(X = 0) = P(X = 1) = .
2 2
"
E(X) = x1p + x2q.
1
EX =
2
D2X = x2p + x2q - (x1p + x2q)2
1 2
= x2p + x2q - x2p2 - 2x1x2pq - x2q2
1 2 1 2
= x2(p - p2) - 2x1x2pq + x2(q - q2)
1 2
= x2pq - 2x1x2pq + x2q = (x1 - x2)2pq.
1 2
1 1
D2X = DX =
4 2
X
1
P(X = xi) = i = 1, 2, . . . , n.
n
"
1
P(X = i) = i = 1, 2, 3, 4, 5, 6.
n
"
x1 + x2 + · · · + xn
EX =
n
x2 + x2 + · · · + x2 x1 + x2 + · · · + xn 2
1 2 n
D2X = -
n n
(n, p)
(n, p)
n
P(X = k) = pkqn-k, k = 0, 1, 2, . . . , n; 0 < p < 1, p + q = 1.
k
"
n
p
n Xi
p
"
n n
n (n - 1)!
EX = k pkqn-k = np pk-1qn-k
k (k - 1)!(n - k)!
k=0 k=1
( i = k - 1)
n-1 n-1
(n - 1)! n - 1
= np piqn-i-1 = np piqn-i-1
i!(n - i - 1)! i
i=0 i=0
= np(p + q)n-1 = np.
m
m
piqm-i = (p + q)m.
i
i=0
n n
n (n - 1)!
EX2 = k2 pkqn-k = np k pk-1qn-k
k (k - 1)!(n - k)!
k=0 k=1
n-1
(n - 1)!
= np (i + 1) piqn-i-1
i!(n - i - 1)!
i=0
n-1 n-1
(n - 1)! (n - 1)!
= np piqn-i-1 + np piqn-i-1
(i - 1)!(n - i - 1)! i!(n - i - 1)!
i=1 i=0
n-2 n-1
(n - 2)! (n - 1)!
= np2(n - 1) plqn-l-2 + np piqn-i-1
l!(n - l - 2)! i!(n - i - 1)!
l=0 i=0
n-2 n-1
n - 2 n - 1
= np2(n - 1) plqn-l-2 + np piqn-i-1
l i
l=0 i=0
= np2(n - 1)(p + q)n-2 + np(p + q)n-1 = np(np + q).
D2X = np(np + q) - (np)2 = npq.


k
P(X = k) = e- k = 0, 1, 2, . . . .
k!
"
n
p
p < 0, 2 n > 20
"
" "
k k-1
EX = ke- = e-
k! (k - 1)!
k=0 k=1
"
l
= e- = e-e = .
l!
l=0
 e
"
xn
ex =
n!
n=0
" " "
k k-1 l
EX2 = k2e- = e- k = e- (l + 1)
k! (k - 1)! l!
k=0 k=1 l=0
" "
l-1 l
= 2e- + e-
(l - 1)! l!
l=1 l=0
" "
i l
= 2e- + e-
i! l!
i=0 l=0
= 2e-e + e-e = 2 + .
D2X = 2 +  - 2 = .
(m, p)
(m, p)
k - 1
P(X = k) = pm(1 - p)k-m, k = m, m + 1, m + 2, . . . ,
m - 1
0 < p < 1 p + q = 1
"
m
m
"
"
k - 1
pmqk-m = 1
m - 1
k=m
(m, p)
"
k - 1
pm+1qk-m-1 = 1
m
k=m+1
(m + 1, p)
k - m = i
m+i-1 m+i-1
=
m-1 i
"
k - 1
EX = k pmqk-m
m - 1
k=m
"
m + i - 1
= (i + m) pmqi
i
i=0
" "
(m + i - 1)! m + i - 1
= pmqi + m pmqi
(i - 1)!(m - 1)! m - 1
i=1 i=0
i = k - m
"
(k - 1)!
EX = pmqk-m + m
(m - 1)!(k + m - 1)!
k=m+1
"
mq (k - 1)!
= pm+1qk-m-1 + m
p m!(k - m - 1)!
k=m+1
"
mq k - 1
= pm+1qk-m-1 + m.
p m
k=m+1
mq m m
EX = + m = (q + p) = .
p p p
EX2
"
k - 1
pm+2qk-m-2 = 1
m + 1
k=m+2
k - m = i
"
m + i - 1
EX2 = (m + i)2 pmqi
m - 1
i=0
" "
m + i - 1 m + i - 1
= m2 pmqi + 2m i pmqi
m - 1 m - 1
i=0 i=0
" "
m + i - 1 m + i - 1
+ i(i - 1) pmqi + i pmqi
m - 1 m - 1
i=0 i=0
 i = k - m
" "
k - 1 k - 1
EX2 = m2 pmqk-m + (2m + 1) (k - m) pmqk-m
m - 1 m - 1
k=m k=m
"
(m + i - 1)!
+ pmqi
(m - 1)!(i - 2)!
i=2
" "
k - 1 k - 1
= m2 + (2m + 1) k pmqk-m - m(2m + 1) pmqk-m
m - 1 m - 1
k=m k=m
"
(k - 1)!
+ pmqk-m
(m - 1)!(k - m - 2)!
k=m+2
"
k - 1
k pmqi = EX
m - 1
k=m
m
EX2 = m2 + (2m + 1) - m(2m + 1)
p
m(m + 1)q2 " k - 1
+ pm+2qk-m-2
p2 m + 1
k=m+2
m m(m + 1)q2
= m2 + (2m + 1) - m(2m + 1) +
p p2
2m2 m m(m + 1)q2
= m2 + + - 2m2 - m +
p p p2
m m(m + 1)q2
= (2m + 1) - m(m + 1) +
p p2
m m(m + 1)
= (2m + 1) + (q2 - p2)
p p2
m m(m + 1)
= (2m + 1) + (q - p)(q + p)
p p2
2
m m(m + 1) m
D2X = (2m + 1) + (q - p) -
p p2 p
m mq
= (2mp + p + mq - mp + q - p - m) =
p2 p2
p
m = 1
"
P(X = k) = p(1 - p)k-1 k = 1, 2, . . . ; 0 < p < 1.
"
"
"
xn | x |< 1
n=1
"
x
xn =
1-x
n=1
"
x
xn = .
1 - x
n=1
"
1
nxn-1 = .
(1 - x)2
n=1
"
1
nxn-1 = ,
(1 - x)2
n=1
"
2
n(n - 1)xn-1 = .
(1 - x)3
n=1
1 - p = q
" "
EX = ipqi-1 = p iqi-1.
i=1 i=1
x = q
1 1
EX = p = .
(1 - q)2 p
" "
EX2 = i2pqi-1 = p [i(i - 1) + i]qi-1
i=1 i=1
" "
= pq i(i - 1)qi-2 + p iqi-1.
i=1 i=1
x = q
2 1 2q 1
EX2 = pq + = + .
(1 - q)3 p p2 p
2
2q 1 1 q
D2X = + - = .
p2 p p p2
Y = X - 1
P(Y = k) = P(X = k + 1) = p(1 - p)k k = 0, 1, 2, . . . ; 0 < p < 1.
1 q
EX = EY - 1 = - 1 = ,
p p
q
D2X = D2Y = .
p2
(N, M)
X
(N, M)
M N-M
k n-k
P(X = k) = , k = 0, 1, 2, . . . , n; M d" N, n d" N.
N
n
"
N M n X
n
"
n M N-M
k n-k
EX = k
N
k=0 n
n
1 M N - M
= k
N
k n - k
n k=1
M M! M(M - 1)! M - 1
k = = = M .
k (k - 1)!(M - k)! (k - 1)!(M - 1 - (k - 1))! k - 1
n
M M - 1 N - M
EX = .
N
k - 1 n - k
n k=1
n
M N - M N
=
k n - k n
k=0
n-1
M M - 1 N - M
EX =
N
j n - j - 1
n j=0
M N - 1 Mn
= = .
N
n - 1 N
n
M
= p EX = np p
N
D2X = E(X2 - X + X) - (EX)2
= E[X(X - 1)] + EX - (EX)2.
EX E[X(X - 1)]
M M(M - 1)(M - 2)!
k(k - 1) = k(k - 1)
k k(k - 1)(k - 2)!M - 2 - (k - 2))!
M - 2
= M(M - 1) .
k - 2
n M N-M
k n-k
E[X(X - 1)] = k(k - 1)
N
k=2 n
n
M(M - 1) M - 2 N - M
=
N
k - 2 n - k
n k=2
n-2
M(M - 1) M - 2 N - 2 - (M - 2)
=
N
j n - 2 - j
n j=0
M(M - 1) N - 2
=
N
n - 2
n
M(M - 1)n(n - 1)
= .
N(N - 1)
M(M - 1)n(n - 1) Mn M2n2
D2X = + -
N(N - 1) N N2
Mn (M - 1)(n - 1)N + N(N - 1) - Mn(N - 1)
=
N N(N - 1)
Mn -MN - nN + N2 + Mn
=
N N(N - 1)
Mn(N - n)(N - M)
= .
N2(N - 1)
M N-M
= p q = 1 - p =
N N
n
1 -
N - n
N
D2X = npq = npq .
1
N - 1 1 -
N
 X f
F X
x
F (x) = f(t)dt x " R
-"
f
X
fX f
X
f
F (x) = f(x).
f
"
f(x)dx = lim F (x) - lim F (x) = 1.
x+" x-"
-"
b a b
P(a d" X < b) = F (b) - F (a) = f(t)dt - f(t)dt = f(t)dt.
-" -" a
P(a d" X < b) = P(a < X d" b) = P(a < X < b) = P(a d" X d" b).
< a, b > x = a x = b P(a d" X < b)
 X fX g
g
X Y = g(X)
fY (y) = fX(h(y)) | h (y) |,
h g
g
g-1 = h
< x, x + "x) < y, y + "y)
P(x d" X < x + "x) = P(y d" Y < y + "y),
y = g(x)
FX(x + "x) - FX(x) = FY (y + "y) - FY (y).
FX(x + "x) - FX(x) FY (y + "y) - FY (y) "y
= .
"x "y "x
g "x 0 "y 0
FX(x + "x) - FX(x) FY (y + "y) - FY (y) "y
lim = lim .
"x0 "x "x0 "y "x
"y
FX(x) = FY (y)
"x
"y
fX(x) = fY (y) .
"x
"x
fY (y) = fX(x) .
"y
x = h(y)
fY (y) = fX(h(y))h (y).
g
P(x d" X < x + "x) = P(y + "y < Y d" y).
fY (y) = fX(h(y))(-h (y)).
 r r = 1, 2, . . . c
X f i = 1, 2, . . .
"
µr(c) = E(X - c)r := (x - c)rf(x)dx,
-"
c = 0
"
mr = EXr := xrf(x)dx
-"
c = m1 c = EX
"
µr = E(X - EX)r := (x - EX)rf(x)dx
-"
X
g(X) g
X f
"
Eg(X) = g(x)f(x)dx,
-"
Eh(X) Eg(X)
E(h(X) + g(X)) = Eh(X) + Eg(X).
 h(X)
g(X)
" " "
| h(x) + g(x) | f(x)dx d" | h(x) | f(x)dx + | g(x) | f(x)dx.
-" -" -"
E(a) = a.
" "
E(a) = af(x)dx = a f(x)dx = a,
-" -"
"
f(x)dx = 1
-"
X
a " R k " N
E(aX)k = akEXk.
k " N
" "
E(aX)k = (ax)kf(x)dx = ak xkf(x)dx = akEXk.
-" -"
"
EX (ax)kf(x)dx
-"
"
= ak xkf(x)dx
-"
E(X) a, b " R
E(aX + b) = aEX + b.
 F
P(X e" x) = 1 - P(X < x) = 1 - F (x) P(X d" x) = P(X < x) = F (x).
x0 X
1 1
F (x0) d" F (x0) e" .
2 2
x0
1
F (x0) = .
2
(a, b)
X (a, b)
1
, x "< a, b >,
b-a
f(x) =
0, x " R- < a, b > .
"
x " (-", a)
x
F (x) = 0dt = 0,
-"
x "< a, b >
a x
1 x - a
F (x) = 0dt + dt = ,
b - a b - a
-" a
 x " (b, ")
a b x
1
F (x) = 0dt + dt + 0dt = 1.
b - a
-" a b
Å„Å‚
ôÅ‚
òÅ‚0, x < a,
x-a
F (x) = , a d" x d" b,
b-a
ôÅ‚
ół1, x > b.
"
b
x b2 - a2 a + b
EX = dx = = ,
b - a 2(b - a) 2
a
b
x2 b3 - a3 a2 + ab + b2
EX2 = = = ,
b - a 3(b - a) 3
a
2
a2 + ab + b2 a + b (b - a)2
D2X = - = .
3 2 12
(a, b)
X (a, b)
Å„Å‚
ôÅ‚ x < 0 (" x > b,
òÅ‚0,
2x
f(x) = , 0 d" x d" a,
ab
ôÅ‚
ół
2x 2
+ , a d" x d" b
ab-b2 b-a
"
x < 0 F (x) = 0 0 d" x d" a
x
2t x2
F (x) = dt = .
ab ab
0
a d" x d" b
a x
2t 2t 2
F (x) = dt + + dt
ab ab - b2 b - a
0 a
a2 x2 a2 2x 2a x2 - a2 2(x - a) a
= + - + - = + + .
ab ab - b2 ab - b2 b - a b - a ab - b2 b - a b
F (x) = 1 x > b
"
a b
2x2 2x2 2x 2a3 2(b3 - a3) b2 - a2
EX = dx + + dx = - +
ab ab - b2 b - a 3ab 3b(b - a) b - a
0 a
a + b
= .
3
a b
2x3 2x3 2x2 a4 b4 - a4 2(b3 - a3)
EX2 = dx + + dx = - +
ab ab - b2 b - a 2ab 2b(b - a) 3(b - a)
0 a
a2 + ab + b2
= .
6
a2 - ab + b2
D2X = .
18


0, x d" 0,
f(x) =
e-x, x > 0.
"
x d" 0 F (x) = 0 x > 0
x
F (x) = e-tdt = 1 - e-x.
0
"
" "
"
1
EX = xe-xdx = - xe-x + e-xdx = ,
0 
0 0
" "
"
EX2 = x2e-xdx = - x2e-x + 2 xe-xdx
0
0 0
"
2 2 2
= xe-xdx = EX = ,
  2
0
2
2 1 1
D2X = - = .
2  2
µ Ã
µ Ã
1 (x - µ)2
"
f(x) = exp - , x " R.
2Ã2
à 2Ą
µ = 0 Ã = 1
1 x2
2
"
Õ(x) = e- , x " R.
2Ä„
"
x
1 t2
2
Åš(x) = " e- dt, x " R.
2Ä„
-"
-x
1 t2
2
Åš(-x) = " e- dt
2Ä„
-"
x
-t = u -1 u2
2
= = " e- dt
-dt = du
2Ä„
"
" x
1 u2 1 u2
2 2
= " e- dt - " e- dt
2Ä„ 2Ä„
-" -"
= 1 - Åš(x).
"
ëÅ‚ öÅ‚
" " " "
x2+y2 y2 x2
2 íÅ‚e- 2 2
e- dx dy = e- dxłł dy
-" -" -" -"
ëÅ‚ öÅ‚2
" " "
y2
x2 x2
2 2 íÅ‚ 2
= e- dy e- dx = e- dxłł
-" -" -"
x = r cos Õ
Õ "< 0, 2Ä„ >, r e" 0.
y = r sin Õ,
ëÅ‚ öÅ‚
" " 2Ä„ "
x2+y2
r2
2 íÅ‚ 2
e- dx dy = e- rdrÅ‚Å‚ dÕ
-" -" 0 0
2Ä„ "
r2
= t
2
= = dÕ e-tdt
rdr = dt
0 0
"
= 2Ä„ -e-t 0 = 2Ä„ lim e-t - 1 = 2Ä„
t"
ëÅ‚ öÅ‚2
"
x2
íÅ‚ 2
e- dxłł = 2Ą.
-"
"
"
x2
2
e- dx = 2Ä„.
-"
Y
"
y2
1
2
EY = " ye- dy = 0,
2Ä„
-"
" "
y2 y2
1 1
2 2
EX2 = " y2e- dy = " y · ye- dy
2Ä„ 2Ä„
-" 0
ëÅ‚ öÅ‚
"
y2
"
1
u = y v = ye- 2 y2 y2
íÅ‚ 2 2
= = " -ye- + e- dyłł
y2
u = 1 v = -e- 2 2Ä„
0
0
"
1 y
"
= - lim + 2Ä„
y2
y"
2Ä„
2
e
"
1 1
"
= - lim + 2Ä„ = 1.
y2
y"
2Ä„
2
ye
D2X = EX2 - (EX)2 = 1.
Y
X = ÃY + µ X
FX(x) = P(X < x) = P(ÃY + µ < x)
x - µ x - µ
= P(Y < ) = Åš( )
à Ã
x-µ
Ã
u-µ
1 t2
= t
Ã
2
= " e- dt =
du
dt =
2Ä„
Ã
-"
x
(u-µ)2
1
= " e- 2Ã2
du.
à 2Ą
-"
X µ Ã
EX = E(ÃY + µ) = ÃEX + µ = µ,
D2X = D(ÃY + µ) = Ã2D2X = Ã2.
µ 
µ 
1 
f(x) = , x " R; µ " R,  > 0.
Ä„ 2 + (x - µ)2
"
x x
1   1 1
F (x) = dt = dt
Ä„ 2 + (t - µ)2 Ä„ 2 1 + t-µ 2

-" -"
x-µ

t-µ
 1 1
= u

= = du
dt = du
Ä„  1 + u2
-"
x-µ

1 1 x - µ 1
= arc tg u = arc tg + .
Ä„ Ä„  2
-"
"
"
x-µ
1 
= t

|x| dx =
dx = dt
Ä„ 2 + (x - µ)2
-"
"
1 |t + µ|
= dx
Ä„ 1 + t2
-"
"
"
|t| 1
dt = ln(1 + t2)
1 + t2 2
0
0
1
= lim ln(1 + t2) - ln 1 = "
t"
2
n
X
Õ(x) Y = X2 Y X
g(x) = x2 h a" g-1
"
1
"
x e" 0 h(y) = y h (y) = y e" 0
2 y
"
1
x < 0 h(y) = - y h (y) = - "
y e" 0
2 y
X
" " y
1 1 1
- " " " 2
Õ(- y) + Õ( y), y > 0, e- , y > 0,
2 y 2 y
2Ä„y
fY (y) = =
0, y d" 0 0, y d" 0.
2 2
n Y = X1 +X2 +
2
· · · + Xn X1, X2, . . . , Xn
n
y
n
1 -1
2
, y > 0,
n y e- 2
n
2
2 “( )
2
fY (y) =
0, y d" 0,
"
“(x) = tx-1e-tdt x > 0
0
EY = n D2Y = 2n
 t n
"
X0 n
2 2 2
"
Tn = Yn = X1 +X2 +· · ·+Xn X0, X1, . . . , Xn
Yn
Tn t n
n+1
n+1
“( )
t2 - 2
2
fT (t) = " 1 + , t " R.
n n
nÄ„“( ) n
2
n
ETn = 0 n = 2, 3, . . . D2Tn = n = 3, 4, . . .
n-2
n = 1 n "
fT Õ Õ
n
F (m, n)
X m Y
nX
n F =
mY
F (m, n) F
Å„Å‚
n
m+n m
“ 2 -1
2
òÅ‚ 2 n x
, x > 0,
m+n
m n m
n
“ “ 2
fF (x) = (x+ )
2 2
m
ół
0, x d" 0.
(&!, F, P) X Y
Z : &! C Z = X + iY
t " R
X
Zt = eitX = cos tX + i sin tX.
X Y
EZ = EX + iEY, E[eitX] = E[cos tX] + iE[sin tX].
X
t
eitX
ÕX(t) = E[eitX].
 X P[X = xi] = pi
i = 1, 2, . . .
"
i
ÕX(t) = eitx pi,
i=1
f(x)
"
ÕX(t) = eitxf(x)dx.
-"
| eitx |= 1 x " R
X P[X = xi] = pi i = 1, 2, . . .
" "
i
| eitx pi |= pi = 1 x " R.
i=1 i=1
X
X f(x)
" "
| eitxf(x) | dx = f(x)dx = 1.
-" -"
t, h " R
| ÕX(t + h) - ÕX(t) | = E[eitX(eihX - 1)]
d" E | eihX - 1 | .
t h
 | ÕX(t) |d" ÕX(0) = 1
ÕX(-t) = ÕX(t) ÕX(-t) = Õ-X(t)
ÕaX+b(t) = eitbÕX(at) a b
ÕX(0) = E(e0) = 1
| ÕX(t) |= E(eitX) d" E | eitX |= 1.
ÕX(-t) = E[cos(-tX) + i sin(-tX)]
= E[cos tX] - iE[sin tX]
= E[cos tX + iE[sin tX] = ÕX(t).
ÕX(-t) = E[e-itX] = E[eit(-X)] = Õ-X(t).
ÕaX+b(t) = E[eit(aX+b)] = eitbE[eitaX] = eitbÕX(at).
a1, a2, . . . , an
X1, X2, . . . , Xn Sn = a1X1 + a2X2 + · · · + anXn
n
ÕS (t) = ÕX (ait)
n i
i=1
X1, X2, . . . , Xn
1 2 n
eita X1, eita X2, . . . , eita Xn
1
ÕS (t) = E eita X1+ita2X2+···+itanXn =
n
1 2 n
= E[eita X1eita X2 . . . eita Xn]
1 2 n
= E[eita X1]E[eita X2] . . . E[eita Xn] =
n
= ÕX (ait).
i
i=1
a1 = a2 = · · · = an = 1
-X
X X
 X ÕX(t) =
Õ-X(t) x " R Õ-X(t) = ÕX(t)
ÕX(t) = ÕX(t) ImÕX(t) = 0
ÕX(t) ÕX(t) = ÕX(t)
Õ-X(t) = ÕX(t) Õ-X(t) =
ÕX(t)
X
P(X = xi) = pi i = 1, 2, . . . E | X |n< "
n e" 1
1
n
| EXn | 1
lim sup = < "
n! R
n"
| t |< R
"
(it)n
ÕX(t) = EXn.
n!
n=0
X
f(x) E | X |n< " n e" 1 r d" n
r X
"
Õ(r)(t) = (ix)reitxf(x)dx,
-"
1
EXr = Õ(r)(0),
ir
n
(it)r (it)n
ÕX(t) = EXr + µn(t),
r! n!
r=0
| µn(t) |d" 3E | X |n µn(t) 0 t 0
k X
ÕX k
Õ(k)(0) = ikEXk.
X
k
X k k
X k
k - 1 X
 ÕX(t)
X F
a < b
R
1 1 e-ita - e-itb
P(a < X < b)+ P(X = a)+ P(X = b) = lim ÕX(t)dt,
2 2 R" it
-R
F a b
R
1 e-ita - e-itb
F (b) - F (a) = lim ÕX(t)dt,
R" 2Ä„ it
-R
"
| ÕX(t) | dt < " X
-"
"
1
f(x) = e-itxÕX(t)dt.
2Ä„
-"
ÕX(t) 2Ä„
X
Ä„
1
pk = P(X = k) = e-itkÕX(t)dt,
2Ä„
-Ä„
k " Z pk
(Fn) F
(Õn) t " R
Õ Õ
F
 (&!, F, P) X : &! R
R g
< 0, ") Eg(X)
Eg(X)
P(| X |e" µ) d" .
g(µ)
µ>0
g g(X)
g
g(X) = g(-X) = g(| X |)
g
X e" µ Ò! g(X) e" g(µ); -X e" µ Ò! g(-X) e" g(µ).
g
-X e" µ Ò! g(X) e" g(µ).
" -µ "
Eg(X) = g(x)f(x)dx e" g(x)f(x)dx + g(x)f(x)dx
-" -" µ
ëÅ‚ öÅ‚
-µ " -µ "
íÅ‚
e" g(µ)f(x)dx + g(µ)f(x)dx = g(µ) f(x)dx + f(x)dxÅ‚Å‚
-" µ -" µ
= g(µ) (P(X d" -µ) + P(X e" µ)) = g(µ)P(| X |e" µ).
" g(x) =| x |p p > 0
E | X |p
P(| X |e" µ) d" ;
µp
µ>0
" g(x) = x2
EX2
P(| X |e" µ) d" .
µ2
µ>0
D2X
P(| X - EX |e" µ) d" .
µ2
µ>0
g
X
g(EX) d" Eg(X).
g
g(x) e" g(x0) + (x - x0).
x0"R  x"R
x = X x0 = EX
g(X) e" g(EX) + (X - EX).
Eg(X) e" Eg(EX) + E(X - EX).
Y t
0 < s < t
1
1
t
s
(E | Y |s) d" E | Y |t
X =| Y |s
g(E | Y |s) d" Eg(| Y |s).
g(x) =| x |r
| E | Y |s|rd" E | Y |sr .
t
r =
s
t
s
| E | Y |s| d" E | Y |t .