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Deductive Logic Probability Theory

π X X X X = X X =

x f z x Physics f Sensor z x f f z

P(X = x) X x X x P(X = x) = 1 /6 P(X = x) P(x) F(x) F(x) = P(X x)

X Y Y Z X Z X X Y X Y X, Y = X Y

P(a) = 1. a P(a, b) = P(a) P(b a) = P(b) P(a b) P(a, b) = P(a b) = P(a b) A B S

P(a, b) = P(a) P(b a) = P(b) P(a b). P(A, B, C) P(B A, C) P(A) P(C A) P(A) P(B A) P(C B) P(A B, C) P(B) P(C A) P(A) P(B) P(C)

P(A, B, C) P(B A, C) P(A) P(C A) P(A) P(B A) P(C B) P(A B, C) P(B) P(C A) P(A) P(B) P(C) P(A, B, C) = P(A B, C) P(B, C) = P(A B, C) P(B C) P(C) = P(A B, C) P(C B) P(B) = P(C B, A) P(B A) P(A)

S H V V P(H V) = 0.95 P(V) = 10 6 P(H, V) = P(H V)P(V) = 0.95 10 6.

b a b P(a i, b j ) = 1 a i P(a i ) = j j P(a i, b j ) x y x y P(x, y) x = 0 x = 1 x = 2 y = 0 0.32 0.03 0.01 y = 1 0.06 0.24 0.02 y = 2 0.02 0.03 0.27

P(a i ) = j P(a i, b j ) P(x, y) x = 0 x = 1 x = 2 y = 0 0.32 0.03 0.01 y = 1 0.06 0.24 0.02 y = 2 0.02 0.03 0.27 P(Y = 0) 0.32. 0.36. 0.40. 0.89.

P(x, y) x = 0 x = 1 x = 2 y = 0 0.32 0.03 0.01 y = 1 0.06 0.24 0.02 y = 2 0.02 0.03 0.27 = 1.0 P(X = 1) = j P(X = 1, y j) = 0.30 P(Y = 0) = i P(x i, Y = 0) = 0.36

P(y, x) = P(y x) P(x) = P(y x) = P(y,x) P(x). P(x, y) x = 0 x = 1 x = 2 y = 0 0.32 0.03 0.01 y = 1 0.06 0.24 0.02 y = 2 0.02 0.03 0.27 P(Y = 0 X = 1) 0.80. 0.19. 0.83. 0.10.

P(x, y) x = 0 x = 1 x = 2 y = 0 0.32 0.03 0.01 y = 1 0.06 0.24 0.02 y = 2 0.02 0.03 0.27 Y = y X = 1 P(y X = 1) = P(X = 1, y) P(X = 1) P(y X = 1) y = 0 0.03/0.30 = 0.1 y = 1 0.24/0.30 = 0.8 y = 2 0.03/0.30 = 0.1 1.0

P(a b) = P(b a) P(a) P(b) P(a b) a P(a) a P(b a) a a L(a) = P(b a) P(b) P(b) = i P(a i, b) = i P(b a i)p(a i ).

f 1,2,3 P(f i w) = P(w f i)p(f i ) = P(w f i)p(f i ) P(w) j P(w f j)p(f j ) i P(w f i ) P(f i ) P(w f i )P(f i ) 10 10 4 2.0 10 4 3 10 4 2.1 10 4 5 10 4 0.5 10 4 P(f 1 w) = 2.0 P(f 2 w) = 2.1 P(f 3 w) = 0.5 4.6 4.6 4.6

A B P(a, b) = P(a) P(b) P(a, b) = P(a b)p(b) P(a, b) = P(a)P(b) P(a)P(b) = P(a b)p(b) P(a b) = P(a) A B

H 1 T 1 H 2 T 2 [ 1 /4 ] [ ] 1/4 1 /2 [ = 1 /2 1/2 ] 1/4 1/4 1/2

P(x, y) x = 0 x = 1 x = 2 y = 0 y = 1 y = 2 P(x, y) x = 0 x = 1 x = 2 y = 0 y = 1 y = 2 y y y x x x

Y D N P(y, d, n) P(y d) = = P(y, d) y P(y, d) n P(y, d, n) P(y, d, n). y n

N B F B F N

N N N P(B, F, N) = P(B) P(F) P(N F, B). B F p(b, F) = p(b) p(f) B F N

P(B, F, N) = P(B) P(F) P(N F, B). P(B N) = b f f P(B) P(F = f) P(N F = f, B) P(B = b) P(F = f) P(N F = f, B = b). P(B) = 0.001, P(F) = 0.1, P(N f, b) = b = b = ( ) f = 0.95 0.8, f = 0.5 0.05 P(B N) = 0.0043,

P(A, B, C, D, E, F) = P(A) P(B) P(C A, B) P(D B) P(E D) P(F D). A B C D E F

P(B) P(B) P(A, B) = P(A)P(B A) = P(B)P(A B) A B A B

P A B B A A B P P A B A P B A P B

A B P P A B A P B A P B

A B P P

B F N N p(b, F N) = p(b N) p(f N) B F N N B F

A E C E B F A B A B C D C D E F E F

X Y D D D X Y X Y X Y D D X Y X Y

E[X] = m(x) = X = µ = i x i P(x i ) X E[X] = 0.48 3 + 0.26 1 + 0.26 0 = 1.7 E E[aX + by] = ae[x] + be[y]

f X A E[f(X) A] = i f(x i )P(x i A) A n E[x n ] = i x n i P(x i ) n E[(x µ) n ] = i (x i µ) n P(x i )

var(x) = E[(x µ) 2 ] = i (x i µ) 2 P(x i ) (X) = σ = (x) X = 1 X = 1 (X) = ( 1 0) 2 1 /2 + ( 1 0) 2 1 /2 = 1 (X) = (1 0.80) 2 0.9 + ( 1 0.80) 2 0.1 = 0.60

0 2 4 6 8 10 0 2 4 6 8 10

A B P(a, b) = P(a)P(b)