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A B PB(A) = P(A)PA(B)
P(B)
(An)nN B
j N, PB(Aj) = P(Aj)PAj (B)+n=0
P(An)PAn(B)
A B P(A B) = P(A)P(B)
P(B)> 0
A
B
PB(A) = P(A) B A
A
B
A B
A
B
A B
(A1,...,An)
(i, j) [[1, n]]2, i=j = P(Ai Aj) = P(Ai)P(Aj)
(A1,...,An)
J
[[1, n]]
P(iJ
Ai) =iJ
P(Ai)
(,P)
E
(,P) E X
E
E= R
X
A E X1(A) {XA}
x E X1({x}) {X=x}
E= R
x R X1(], x]) {X x}
X {X=x} x X()
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X : E X PX :
X() [0, 1]
x P(X=x) A P(X()) PX(A) = P(XA) PX X()
X
A P(X()) P(A) > 0 X
A
PX|A:
X() [0, 1]
x PA(X=x)
X X X U(X())
x X(),P(X=x) = 1
card X()
X X U([[1, 6]])
X {0, 1} X
p X B(p) P(X= 1) = p P(X= 0) = 1 p
X : [[0, n]] X n p X B(n, p) k [[0, n]] P(X=k) =
nkp
k(1 p)nk B(n, p) n p X
E
N
N1= N p N2= N N1=N(1 p)
n
X
n
P(X=k) =
Npk
NNpnk
Nn
(, A,P)
X
(, A)
X()
X() A
X
(, A,P)X() X() ={xn}nI I N X (pn)nI n I pn= P(X=xn)
n I , pn [0, 1] nI
pn= 1
X
(, A) X() =
{xn}nI {pn}nI n I, pn [0, 1] nI
pn = 1 P
(, A) n I , P(X=xn) =pn
FX
X
(, A)
R x R, FX(x) = P(X x)
FX limx
FX(x) = 0 limx+
FX(x) = 1
X Y (, A) (X, Y) (X(), Y())
(, A)
(xi, yj) (X(), Y()) ((X, Y) = (xi, yj)) { |X() =xi Y() =yj}
(X=xi, Y =yj)
(X, Y)
(X, Y)
P(X=xi, Y =yj) (xi, yj) X() Y()
X
Y
(X, Y)
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x X() P(X=x) =
yY()
P(X=x, Y =y)
y Y() P(Y =y) =
xX()
P(X=x, Y =y)
x X() P(X=x) > 0 Y X=x Px Y()
yj Y() P
x(yj) =PX=x(yj) =
P(X=x, Y =yj)
P(X=x)
X Y (, A,P) (X, Y)
(xi, yj) X() Y(), P(X=xi, Y =yj) = P(X=xi)P(Y =yj)
(X, Y) A X()
B Y() P(XA, Y B) = P(XA) P(Y B)
(X, Y) f g
f(X)
g(Y)
(X1,...,Xn) (, A,P)
(x1,...,xn) X1() ... Xn(), P(X1= x1,...,Xn= xn) = P(X1= x1) ...
P(Xn= xn)
(X1,...,Xn)
(A1,...,An) P(X1()) ...P(Xn()) (Xi Ai)i[[1,n]]
(X1,...,Xn)
(f1,...,fn) n (X1,...,Xn)
(f1(X1),...,fn(Xn))
(Xn)nN (, A,P) (Xn)nN
(Xn)nN (, A,P) (Xn)nN
(i, j) N2, i=j =Xi Xj
(Xn)nN
(X0, X1,...,Xn)
(Xn)nN
Xi
B(12)
