4 4. A p X A 1 X X A 1 A 4.3 X p X p X S(X) = E ((X p) ) X = X E(X) = E(X) p p 4.3p < p < 1 X X p f(i) = P (X = i) = p(1 p) i 1, i = 1,, r + r
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1 X P (X = 1) =.4, P (X = ) =.3, P (X = 1) =., P (X = ) =.1 E(X) = = 4. X Y = X P (X = ) = P (X = 1) = P (X = ) = P (X = 1) = P (X = ) =. Y P (Y = ) = P (X = ) =., P (Y = 1) = P (X = 1) + P (X = 1) =.4 P (Y = 4) = P (X = ) + P (X = ) =.4 E(Y ) = = E(Y ) = P (Y = ) + 1 P (Y = 1) + 4 P (Y = 4) (4.1) = 1 P (X = 1) + ( 1) P (X = 1) + P (X = ) + ( ) P (X = ) = k P (X = k) k= Y = g(x) E(Y ) = k g(k)p (X = k) (4.) E(X ) (E(X)) 4.1 X Z = max(x 1, ) (1) Z () Z (3) Z f X () =., f X (1) =.3, f X () =., f X (3) =.15, f X (4) =.1, f X (5) =.5
2 4 4. A p X A 1 X X A 1 A 4.3 X p X p X S(X) = E ((X p) ) X = X E(X) = E(X) p p 4.3p < p < 1 X X p f(i) = P (X = i) = p(1 p) i 1, i = 1,, r + r + r 3 + = (1 r) 1 f(1) + f() + = 1 i E(X) = i p(1 p) i 1 = p(1 p) i 1 i=1 i=1 j=1 = p(1 p) j 1 (1 p) i j = (1 p) j 1 = 1 p j=1 i=j j=1 4.4 p < p < 1 n X X n, p f(i) = P (X = i) = ( ) n p i (1 p) n i, i i =, 1,,..., n 4.5 a X X a f(i) = P (X = i) = ai i! e a, i =, 1,, = p X X 31536, p X (= p) 1 1
3 4 3 1, p X c { c(1 x ), 1 x 1 f 1 (x) =, { c( x ), x f (x) =, 1 c f 1 (x) c 1 1 ( (1 x )dx = c ) = 1 c = E(X) = x(1 x )dx = f (x) c ( x )dx = c ( x)dx + c ( + x)dx = c (4 ) + c (4 ) = 4c c = 1 4 E(X) = 1 x( x )dx = x x dx = 4.5 X X E(X), x < F 1 (x) = x x, x 1 1, x > 1, x < F (x) = x, x 1 1, x > 1 [, 1] f 1 (x) = x (4.3) 1 xf 1 (x)dx = 1 = 1 (4.4)
4 4 4 F (x) 1 f (x) = 1 x xf (x)dx = 1 1 xdx = 1 3 (4.5) (4.6) 4.6 X { 1 e x, x F (x) =, x < f(x) = d dx F (x) = { e x, x, x < E(X) = x e x dx = [ xe x] + e x dx = 1 X ( x ) E(X) = xf(x)dx = du f(x)dx = = (1 F (u))du ( u ) f(x)dx du E(X) = e x dx = 1 4.7X µ, σ z = x µ σ E(X) = E(X) = = σ x 1 σ e (x µ) /(σ ) dx 1 (σz + µ) e z / dz z 1 e z / dz + µ 1 e z / dz ze z / 1 1 E(X) = µ
5 4 5 V (X) = E ( (X µ) ) = = σ (x µ) 1 e (x µ) /(σ ) dx, σ = σ = σ z 1 e z / dz V (X) = σ z 1 e z / dz + σ z 1 e z / dz z 1 e z / dz = 1 [ ze z / ] ( z = x µ ) σ z 1 e z / dz + 1 e z / dz = 1 4.8X µ, σ Z = X µ σ Z Z ( X µ P (Z x) = P σ = σx+µ ) x = P (x σx + µ) (4.7) 1 σ e (u µ) /(σ ) du d dx P (Z x) = σ 1 σ e x / = 1 e x / (4.8) 1 X (X E(X))/ S(X) 1 4.9, 1 X Z = max{x, } Z x 1 e x / dx = 1 [ e x / ] = X [a, b] (a, ), (c, d), (b, ) a < c < b, d > )X (a + b + c)/3 4.1 Y [ 1, 1] Z = Y + (1) Y f Y (x) Z f Z (x) () Y (3) Z
6 4 6 (1) 1 {.5, 1 x 1 f Y (x) =, Z Y Y {.5, x 4 f Z (x) =, P (Z x) = P (Y + x) = P (Y x ) 1 x ()(3) f Z (x) = d d ( P (Z x) = dx dx P Y x ) 1 = 1 f Y (x/ 1) = 1 4 E(Y ) = E(Z) = x dx = x dx 4 = 16 8 = Z = Y + f Z (x) = f Y (x) + Z [ 1, 1] U [, 1] X X = au + b X 4.8 X [a, b] 4.9 A p 4.1 A B (1) () (1) () (3)
7 m a m + 1 b k X f(x) (1) k X, a, m, k () k X, b, m, k (3) k Z Z X, a, b, m, k (4) Z f(x) (5) g(x) Z g(x) k, k + 1,..., m a m + 1, m +,..., k + X +b m + 1 a X m k (1) X m k a a a(min {X, m k} + 1) () X m k + 1 b(k + X m) k b max {X + k m, } (3) (4) E(Z) = b Z = b max {X + k m, } a(min {X, m k} + 1) i m k m k (i + k m)f(i) a if(i) a a(m k) i= i>m k (5) A Y Y A = k X f(i) P (X = i A = k) = P (Y = k + i) P (Y k) = g(k + i) 1 G(k 1) = f(i) G(k) = g() + g(1) + + g(k) (4) m = 64 k = 63 X = 1 X = 1 X = 3 X =... min {X, m k} + 1(= A) max {X + k m, } (= B) m = 64, k = 63, X = 1 A =, B = X = A =, B = 1 k k X k Y X k = n Y k Y = n + k
8 4 8 g(i) = f(i k) X 1,.5 Y = 14 + (X 5) Y Y 14,5 13,5 max{y m, } m g(m) Excel g(m) 1398 m 14 (1) Y µ s µ 3s, µ + 3s k ; k 1 () Excel =binomdist(x,n,p,false) n, p P (Y = i) k i k 1 (3) g(m) g(m) = E(max{Y m, }) = g(m) k 1 i=k max{i m, }P (Y = i) k k 1 (1) X X () X (3) Excel (4) x 5 + 1(x 5) X F (x) f(x)
9 4 9 [, ) m m C(m) C(m) 7 min{x, m} 5 max{m X, } (1) C(m) g(m) g(m) f(x) () g(m) m g(m) (3) g(m) m 4.1 X X n = 6, p (1) X/n () p =.1 X/n p /p 1% n = 6 (1) X n, p np np(1 p) E ( ) X = p, V n ( ) X p(1 p) = n n () X 1% X/ X X ( ) X/n p P.1 = P (.9p Xn ) p 1.1p = P (.9np X 1.1np) Excel BINOMDIST(...,TRUE) =BINOMDIST(66,6,.1,TRUE) - BINOMDIST(53,6,.1,TRUE).64 9% 11%.6 % 8% 1%.1. 66, 537, n X X n, p n Z = X/n Z n = 1 p =.1,.,.3 Z p > {a, a 1,...} G(z) = a i z i (4.9) G() = a 1 z = G () = a 1 z = G () = a n z = i= G (n) () = n!a n a n = G(n) (), n =, 1,,... (4.1) n!
10 4 1 generating function X {P (X = i), i =, 1,,...} G X (z) = P (X = i) z i = E(z X ) i= z i P (X = i) z X 4.13 p G(z) = z (1 p) + z 1 p = 1 p + pz (4.11) z P (X = ) = 1 p, P (X = 1) = p (4.1) 4.14 n, p G(z) = P (X = i) z i = i= n i= ( ) n p i (1 p) n i z i (4.13) i = (pz + 1 p) n (4.14) G (z) = np((pz + 1 p) n 1 P (X = 1) = np(1 p) n 1 (4.15) G (z) = n(n 1)p (pz + 1 p) n n(n 1) P (X = ) = p (1 p) n (4.16) G (z) =... (4.17) X {p i } Y {q i } i p i = q i X Y X G X (z) Y G Y (z) z z i X, Y Z X
11 X, Y p X + Y, p p 1 p + pz X + Y E(z X+Y ) = E(z X z Y ) = E(z X )E(z Y ) (4.18) = (1 p + pz) = (1 p) + p(1 p)z + p z (4.19) X, Y z i (1 p), p(1 p), p, p X + Y, p, p 4.16 X n, p Y m, p X + Y n + m, p n, p (pz +1 p) n X +Y G X+Y (z) X, Y G X+Y (z) = E(z X+Y ) = E(z X z Y ) = E(z X )E(z Y ) = G X (z)g Y (z) = (pz + 1 p) n (pz + 1 p) m = (pz + 1 p) n+m n + m, p X + Y n + m, p X + Y P (X + Y = k) = i P (X + Y = k Y = i)p (Y = i) = = min{k,m} i=max{,k n} min{k,m} i=max{,k n} P (X = k i)p (Y = i) min{k,m} = p k (1 p) n+m k ( ) ( ) n m p k i (1 p) n k+i p i (1 p) m i k i i i=max{,k n} n!m! (k i)!(n k + i)!i!(m i)! ( ) n+m k d dz G X(z) = d dz E ( z X) = E ( d dz zx ) = E(Xz X 1 ) (4.)
12 4 1 z = 1 E(X) z = 1 E(X(X 1)) d k dz k G X(z) = E(X(X 1) (X k + 1)) (4.1) z=1 k 4.17 X p p G(z) = p(1 p) i 1 z i = i=1 pz 1 (1 p)z (4.) G (z) = p(1 (1 p)z) + (1 p)pz p (1 (1 p)z) = (1 (1 p)z) (4.3) E(X) = G (1) = 1 p (4.4) G (z) = p(1 p) (1 (1 p)z) 3 (4.5) E(X(X 1)) = G (1) = V (X) = E(X(X 1)) + E(X) (E(X)) = (1 p) p (4.6) (1 p) p + 1 p 1 p = 1 p p (4.7) i=1 i ix i 1 = x i 1 = x i 1 = i=1 i=1 j=1 i=1 j=1 j=1 i=j j=1 i=j j=1 i i(i + 1)x i 1 = jx i 1 = j x i 1 = E(X) = ip(1 p) i 1 = 1 p i=1 i=1 x j 1 1 x = 1 (1 x) (4.8) j=1 E(X ) = i(i + 1)p(1 p) i 1 ip(1 p) i 1 = p 1 p i=1 j xj 1 1 x = (1 x) 3 (4.9) (4.3) (4.31) V (X) = p 1 p 1 p = 1 p p (4.3)
13 4 13 X e θx M X (θ) e M X (θ) = E ( θa i P (X = a i ), X e θx) i = (4.33) e θx f X (x)dx, X e θ = z d k dθ k M X(θ) = E(X k ) (4.34) θ= X F (x) X F (x) 4.18 X 1 E ( e θx) = e θx 1 e x / dx = e θ / 1 e (x θ) / dx = e θ / (4.35) θ M X(θ) = θe θ / M X(θ) = e θ / + θ e θ / (4.36) (4.37) E(X) = M X() = (4.38) V (X) = M X() M X() = 1 (4.39) 4.19 µ, σ X µ σ X
14 4 14 Z X = σz + µ X M X (θ) = E(e θ(σz+µ) ) = E(e θσz )E(e θµ ) = e θµ M Z (θσ) = exp (θµ + σ θ ) (4.4) M X(θ) = ( µ + σ θ ) exp (θµ + σ θ ) ) (4.41) ( M X(θ) = σ + ( µ + σ θ ) ) exp (θµ + σ θ ) (4.4) E(X) = M X() = µ (4.43) V (X) = M X() M X() = σ (4.44) 4. X µ, σ Y ν, τ X + Y µ + ν, σ + τ X, Y M X (θ) = exp (θµ + σ θ ) M Y (θ) = exp (θν + τ θ ) (4.45) (4.46) e θx, e θy X + Y M X+Y (θ) = M X (θ)m Y (θ) = exp (θµ + σ θ ) exp (θν + τ θ ) = exp (θ(µ + ν) + (σ + τ )θ ) (4.47) (4.48) µ + ν, σ + τ X + Y µ + ν, σ + τ 4.17 σ X X σ σ 4
A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P
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統計学のポイント整理
.. September 17, 2012 1 / 55 n! = n (n 1) (n 2) 1 0! = 1 10! = 10 9 8 1 = 3628800 n k np k np k = n! (n k)! (1) 5 3 5 P 3 = 5! = 5 4 3 = 60 (5 3)! n k n C k nc k = npk k! = n! k!(n k)! (2) 5 3 5C 3 = 5!
応用数学III-4.ppt
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確率論と統計学の資料
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IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
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