確率・統計勉強会2

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ୈ 2 ճ ֬཰ɾ౷ܭͷجૅษڧձ Hiroaki Tanaka Augmented Human Communication Laboratory, Department of Information Schience, Nara Institute of Science and Technology December 14, 2016 1 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ 1 લճͷ෮श 2 ֬཰ม਺ͷಠཱੑ 3 ֬཰෼෍ͷಛੑ஋ 4 ੵ཰฼ؔ਺ 5 ۙࣅ๏ଇ 6 ࣍ճ༧ࠂ 2 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ લճͷษڧձͰͷϙΠϯτ • ֬཰͸ू߹্ʹఆٛ͞ΕΔ • ֬཰ม਺͸ؔ਺͚ͩͲɼม਺ͱݟͯѻͬͯ΋໰୊ͳ͍ • ֬཰ม਺ͷΠϝʔδ͸ɼ ʮऔΓಘΔ஋ʹ֬཰Λ൐ͬͨม਺ʯ 3 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ 1 લճͷ෮श 2 ֬཰ม਺ͷಠཱੑ 3 ֬཰෼෍ͷಛੑ஋ 4 ੵ཰฼ؔ਺ 5 ۙࣅ๏ଇ 6 ࣍ճ༧ࠂ 4 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ֬཰ม਺ͷಠཱੑ • ෳ਺ͷ֬཰ม਺Λߟ͑Δࡍʹɼͦͷ֬཰ม਺ؒͷ “ؔ܎”͕ॏཁͱͳΔ • “ಠཱੑ”͸ͦͷؔ܎ͷதͰ΋࠷΋ॏཁͳ΋ͷ ֬཰ม਺ͷಠཱ ֬཰ม਺ X1 , . . . , Xn ʹର͠ɼ FX1···Xn (x1 , . . . , xn ) = n ∏ i=1 FXi (xi ), ∀x1 , . . . , xn ∈ R ͕੒Γཱͭͱ͖ɼX1 , . . . , Xn ͸ޓ͍ʹಠཱͱ͍͏ɽ 5 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ಠཱੑͱ෼෍͕͍࣋ͬͯΔ৘ใ • ֬཰ม਺ͷಠཱੑ͸ɼ݁ہ͸ಉ࣌෼෍͕पล෼෍ͷੵͰॻ͚Δͱ͍͏ ͜ͱ • ಉ࣌෼෍͕Θ͔Ε͹શͯͷपล෼෍͸෼͔Δɿ FX (x) = lim t→∞ FXY (x, t) • Ұൠʹٯ͸੒Γཱͨͳ͍͕ɼಠཱੑ͕͋Ε͹ɼͦͷٯ͕੒Γཱͭ • ޙͰѻ͏౷ܭతਪଌཧ࿦͸ɼجຊతʹಠཱੑΛԾఆͯٞ͠࿦͢Δ 6 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ p.m.f. ͱ p.d.f. Ͱॻ͘ಠཱੑ • ֬཰ม਺ͷಠཱੑͷఆٛ͸ɼҎԼͷ΋ͷͱಉ஋ɿ ֬཰ม਺ͷಠཱੑ ֬཰ม਺ X1 , . . . , Xn ͕ಠཱͰ͋Δ͜ͱͱҎԼ͸ಉ஋Ͱ͋Δɿ D-type P(X1 = x1 , . . . , Xn = xn ) = n ∏ i=1 P(Xi = xi ), ∀(x1 , . . . , xn ) ∈ E C-type fX1···Xn (x1 , . . . , xn ) = n ∏ i=1 fXi (xi ), ∀(x1 , . . . , xn ) ∈ Rn 7 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ಠཱͳ֬཰ม਺ͷ࿨ͷ෼෍ • ಠཱͳ֬཰ม਺ͷ࿨ͱͯ͠ಘΒΕΔ “֬཰ม਺”ͷ෼෍ʹ͍ͭͯߟ࡯ ͢Δ • ౷ܭతਪଌͰ༻͍ΒΕΔਪఆྔ͸ɼಠཱͳ֬཰ม਺ͷ࿨ͱͯ͠ද͞Ε Δ΋ͷ͕গͳ͘ͳ͍ • ٞ࿦Λ؆୯ʹ͢ΔͨΊʹɼ֬཰ม਺͕̎ͭͷ৔߹Λߟ͑Δ 8 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ D-type ֬཰ม਺ͷ࿨ͷ෼෍ • (X, Y ) ڞʹ D-type ֬཰ม਺ͱ͠ɼX ∥ Y ͱ͢Δ • pX Λ X ͷ p.m.f.ɼpY Λ Y ͷ p.m.f. ͱ͠ɼVX = { x1 , x2 , . . . } Λ X ͷऔΓಘΔ஋ͷू߹ɼVY = { y1 , y2 , . . . } Λ Y ͷऔΓಘΔ஋ͷू ߹ͱ͢Δ • Z = X + Y ͸ D-type ֬཰ม਺Ͱ͋Δ͕ɼͦͷ෼෍͸ҎԼͷΑ͏ʹ ͯ͠ٻ·Δɿ pZ (z) =P(X + Y = z) = ∑ x∈VX P(X = x, Y = z − x) = ∑ x∈VX pXY (x, z − x) = ∑ x∈VX pX (x)pY (z − x) ৞ΈࠐΈ , z − x ∈ VY . 9 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ D-type ֬཰ม਺ͷ࿨ͷ෼෍ ྫ ϙΞιϯ෼෍ X1 ∼ Po(λ1 ), X2 ∼ Po(λ2 ), X1 ∥ X2 ͱ͢Δͱɼ X1 + X2 ∼ Po(λ1 + λ2 ) X1 + X2 = Z ͱ͢ΔͱɼZ ΋ඇෛ੔਺஋ΛऔΔ D-type ֬཰ม਺Ͱ͋Δɽ ∀z ∈ Z+, pZ (z) =P(Z = z) =P(X1 + X2 = z) = ∞ ∑ x=0 pX1 (x)pX2 (z − x) = z ∑ x=0 pX1 (x)pX2 (z − x) (∵ z − x ∈ Z+) 10 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ D-type ֬཰ม਺ͷ࿨ͷ෼෍ ྫ = z ∑ x=0 exp(−λ1 ) λx 1 x! exp(−λ2 ) λz−x 2 x!(z − x)! = exp {−(λ1 + λ2 )} z ∑ i=0 λx 1 λz−x 2 x!(z − x)! = exp {−(λ1 + λ2 )} 1 z! z ∑ i=0 z! x!(z − x)! λx 1 λz−x 2 = exp {−(λ1 + λ2 )} (λ1 + λ2 )z z! (∵ ೋ߲ఆཧ) ͱͳΓɼ͜Ε͸ Po(λ1 + λ2 ) ͷ p.m.f ͱͳ͍ͬͯΔɽ 11 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ C-type ֬཰ม਺ͷ࿨ͷ෼෍ • X, Y Λڞʹ C-type ֬཰ม਺ͱ͠ɼX ∥ Y ͱ͢Δ • Z = X + Y ͸ C-type ֬཰ม਺ͱͳΓɼͦͷ j.p.d. ͸ҎԼͷΑ͏ʹ͠ ͯٻ·Δɿ FZ (z) =P(Z ≤ z) =P(X + Y ≤ z) = ∫∫ x+y≤z fXY (x, y) dxdy = ∫ ∞ −∞ ∫ z−y −∞ fXY (x, y) dxdy = ∫ ∞ −∞ {∫ z−y −∞ fX (x) dy } fY (y) dy = ∫ ∞ −∞ FX (z − y)fY (y) dy = ∫ ∞ −∞ FY (z − x)fX (x) dx 12 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ C-type ֬཰ม਺ͷ࿨ͷ෼෍ ద౰ͳਖ਼ଇ৚݅ͷ΋ͱͰɼZ ͷ p.d.f. fZ ͸ fZ (z) = d dz FZ (z) = d dz ∫ ∞ −∞ FX (z − y)fY (y) dy = ∫ ∞ −∞ d dz FX (z − y)fY (y) dy = ∫ ∞ −∞ fX (z − y)fY (y) dy = ∫ ∞ −∞ fY (z − x)fX (x) dx ৞ΈࠐΈ 13 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ C-type ֬཰ม਺ͷ࿨ͷ෼෍ ྫ ਖ਼ن෼෍ X ∼ N ( µ1 , σ2 ) , Y ∼ N ( µ2 , σ2 ) , X ∥ Y ͱ͢Δͱɼ X + Y ∼ N ( µ1 + µ2 , σ2 1 + σ2 2 ) ͕੒Γཱͭɽ Z = X + Y ͱ͢ΔͱɼZ ͷ c.d.f. ͸ FZ (z) =P(Z ≤ z) =P(X + Y ≤ z) = ∫∫ x+y≤z fXY (x, y) dxdy = ∫ ∞ −∞ ∫ z−y −∞ fXY (x, y) dxdy 14 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ C-type ֬཰ม਺ͷ࿨ͷ෼෍ ྫ = ∫ ∞ −∞ {∫ z−y −∞ fX (x) dx } fY (y) dy (∵ X ∥ Y ) = ∫ ∞ −∞ FX (z − y)fY (y) dy ͱͳΔɽΑͬͯɼ fZ (z) = d dz FZ (z) = ∫ ∞ −∞ fY (z − x)fX (x) dx = ∫ ∞ −∞ 1 √ 2πσ2 1 exp [ − (z − x − µ1 )2 2σ2 1 ] 1 √ 2πσ2 2 exp [ − (x − µ2 )2 2σ2 2 ] dx = ∫ ∞ −∞ 1 2π √ σ2 1 σ2 2 exp [ − 1 2σ2 1 σ2 2 { σ2 2 (z − x − µ1 )2 + σ2 1 (x − µ2 )2 } ] dx 15 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ C-type ֬཰ม਺ͷ࿨ͷ෼෍ ྫ exp ͷ {} ಺ΛऔΓग़ͯ͠ɼ σ2 2 (z − x − µ1 )2 + σ2 1 (x − µ2 )2 =σ2 1 ( x − µ2 )2 + σ2 2 (z −x + µ2 − µ2 − µ1 )2 =(σ2 1 + σ2 2 ) { x − µ2 − σ2 2 σ2 1 + σ2 2 (z − µ2 − µ1 ) }2 + ( σ2 2 − σ4 2 σ2 1 + σ2 2 ) (z − µ2 − µ1 )2 Ͱ͋Δ͔Βɼ fZ (z) = 1 2π √ σ2 1 σ2 2 exp [ − {z − (µ2 + µ1 )}2 σ2 1 + σ2 2 ] × ∫ ∞ −∞ exp [ − σ2 1 + σ2 2 2σ2 1 σ2 2 { x − µ2 − σ2 2 σ2 1 + σ2 2 (z − µ2 − µ1 ) }2 ] dx 16 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ C-type ֬཰ม਺ͷ࿨ͷ෼෍ ྫ exp ͷ {} ಺ΛऔΓग़ͯ͠ɼ σ2 2 (z − x − µ1 )2 + σ2 1 (x − µ2 )2 =σ2 1 ( x − µ2 )2 + σ2 2 (z −x + µ2 − µ2 − µ1 )2 =(σ2 1 + σ2 2 ) { x − µ2 − σ2 2 σ2 1 + σ2 2 (z − µ2 − µ1 ) }2 + ( σ2 2 − σ4 2 σ2 1 + σ2 2 ) (z − µ2 − µ1 )2 † ਖ਼ن෼෍ͷܭࢉͰ͸ɼࢦ਺෦෼Λฏํ׬੒ͯ͠Ψ΢εੵ෼ͷܗ΁ Ͱ͋Δ͔Βɼ fZ (z) = 1 2π √ σ2 1 σ2 2 exp [ − {z − (µ2 + µ1 )}2 σ2 1 + σ2 2 ] × ∫ ∞ −∞ exp [ − σ2 1 + σ2 2 2σ2 1 σ2 2 { x − µ2 − σ2 2 σ2 1 + σ2 2 (z − µ2 − µ1 ) }2 ] dx 16 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ C-type ֬཰ม਺ͷ࿨ͷ෼෍ ྫ ͜͜Ͱɼ ∫ ∞ −∞ exp [ − σ2 1 + σ2 2 2σ2 1 σ2 2 { x − µ2 − σ2 2 σ2 1 + σ2 2 (z − µ2 − µ1 ) }2 ] dx = √ 2πσ2 1 σ2 2 σ2 1 + σ2 2 ( Ұൠతʹɼ ∫ ∞ 0 exp(−ax2) dx = 1 2 √ π a ) Ͱ͋Δ͔Βɼ fZ (z) = 1 √ 2π (σ2 1 + σ2 2 ) exp [ − {z − (µ2 + µ1 )}2 σ2 1 + σ2 2 ] ͱͳΔɽ͕ͨͬͯ͠ɼZ ∼ N ( µ1 + µ2 , σ2 1 + σ2 2 ) Ͱ͋Δɽ 17 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ γϛϡϨʔγϣϯ • x(1) 1 , . . . , x(1) 1000 iid ∼ N ( 1, 12 ) , x(2) 1 , . . . , x(2) 1000 iid ∼ N ( −1, 12 ) ͔Β x(1) 1 + x(2) 1 , . . . , x(1) 1000 + x(2) 1000 Λܭࢉ͠࿨ͷ෼෍͕ N ( 1 − 1, 12 + 12 ) = N (0, 2) ʹͳΔ͜ͱΛ͔֬ΊΔ Figure: x(1) i + x(2) i , i = 1, . . . , 1000 ͷώετάϥϜͱີ౓ਪఆͷ݁Ռ 18 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ෼෍ͷ࠶ੜੑ • ಠཱʹಉҰͷ෼෍ʹै͏֬཰ม਺ͷ࿨ͷ෼෍͕ɼݩͷ֬཰ม਺ͱಉ͡ ෼෍ʹͳΔੑ࣭Λɼ෼෍ͷ࠶ੜੑͱ͍͏ • ઌड़ͷྫͰݴ͑͹ɼϙΞιϯ෼෍΍ਖ਼ن෼෍͸࠶ੜੑΛ࣋ͭ • ଞʹ΋ɼίʔγʔ෼෍ɾΨϯϚ෼෍ɾೋ߲෼෍ɾΧΠೋ৐෼෍ͳͲ͸ ࠶ੜੑΛ࣋ͭ 19 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ 1 લճͷ෮श 2 ֬཰ม਺ͷಠཱੑ 3 ֬཰෼෍ͷಛੑ஋ 4 ੵ཰฼ؔ਺ 5 ۙࣅ๏ଇ 6 ࣍ճ༧ࠂ 20 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ֬཰෼෍ͷಛੑ஋ • ֬཰ม਺͕ͲΜͳ஋ (ࣄ৅) ΛͲΜͳ֬཰ͰऔΔͷ͔ʁΛنఆͨ͠΋ ͷ͕ “֬཰෼෍” • ݱ࣮ͷෆ࣮֬ੑΛɼ֬཰෼෍ͱ͍͏ܗͰͱΒ͍͑ͯΔ • ͔͠͠ɼݱ࣮ʹ͸σʔλ͕ઌʹ༩͑ΒΕ͍ͯͯ֬཰෼෍͸ະ஌ • σʔλ͔Β֬཰෼෍͕͖ͪΜͱ෼͔Ε͹Կ΋໰୊ͳ͍͕ɼຆͲͷ৔߹ ͸෼͔Βͳ͍ • ͦͷΑ͏ͳ৔߹ɼ֬཰෼෍ͷಛੑ஋͚ͩͰ΋σʔλ͔Β໌Β͔ʹ͠ ͍ͨ 21 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ظ଴஋ (1 ม਺) ظ଴஋ (1 ม਺) Մଌؔ਺ ϕ : R → R ʹର͠ɼ E [φ(X)] = ∫ φ(x) dF(x) =        ∞ ∑ i=1 φ(ai )pi X : D-type ∫ R φ(x)fX (x) dx X : C-type ͕༗ݶ֬ఆͷͱ͖ɼ ͜ΕΛ φ(X) ͷظ଴஋ͱ͍͏ɽಛʹɼ φ(X) = X ͷͱ͖ E [X] Λ X ͷظ଴஋ͱ͍͏ɽ • ֬཰෼෍ͷಛੑ஋ͷதͰ࠷΋ॏཁ • ฏۉͱ΋ݺ͹ΕΔ • ֬཰෼෍ͷॏ৺ɾͭΓ͍͋఺ͱ͍ͬͨղऍʹͳΔ 22 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ظ଴஋ܭࢉͷྫ ϕϧψʔΠ෼෍ͷظ଴஋ X ∼ B(1, p) ͷͱ͖ɼE [X] ΛٻΊΑɽ X ͷऔΓ͏Δ஋ͷू߹͸ { 0, 1 } ͳͷͰɼ E [X] =0 × P(X = 1) + 1 × P(X = 1) =p. 23 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ظ଴஋ (ଟม਺) • X1 , . . . , Xn ͷظ଴஋Λߟ͑Δ • 1 ม਺ͷ৔߹Λ୯७ʹ֦ு͢Δ ظ଴஋ (ଟม਺) X1 , . . . , Xn ͕ C-type ͷͱ͖ɼX ͷظ଴஋Λ E [φ (X))] = ∫ R · · · ∫ R φ(x1 , . . . , xn )fX(x1 , . . . , xn ) dx1 · · · dxn ͰఆΊΔɽ • X ͕ D-type ͷͱ͖΋ɼಉ༷ͷ֦ுʹΑͬͯظ଴஋Λఆٛ͢Δ 24 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ෼ࢄ Ϟʔϝϯτͱ෼ࢄ • φ(x) = xr ͷͱ͖ɼE [φ(X)] Λ X ͷݪ఺पΓͷ r ࣍Ϟʔϝ ϯτ·ͨ͸෼෍ F ͷ r ࣍Ϟʔϝϯτͱ͍͏ • φ(x) = (x − E [X])r ͷͱ͖ͷ E [φ(X)] Λظ଴஋ (ฏۉ) पΓ ͷ r ࣍Ϟʔϝϯτͱ͍͏ • ಛʹɼظ଴஋पΓͷ 2 ࣍ϞʔϝϯτΛ෼ࢄͱ͍͍ V [X] ͱ ॻ͘ʀ V [X] = ∫ R (x − E [X])2 dF(x). • ෼ࢄ͸ “Ͳͷఔ౓෼෍͕޿͕͍ͬͯΔ͔” ΛଌΔई౓ 25 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ظ଴஋ͱ෼ࢄͷੑ࣭ ظ଴஋ͱ෼ࢄ͕ຬͨ͢ੑ࣭ E [ X2 ] < ∞, a, b : ఆ਺ͱ͢ΔͱɼҎԼ͕੒Γཱͭɿ E [aX + b] = aE [X] + b, V [X] = E [ X2 ] − (E [X])2 , V [aX + b] = a2V [X] . • ظ଴஋ʹ͸ઢܗੑ͕͋Δ • ෼ࢄ͸εέʔϧม׵͸ 2 ৐ΦʔμʔͰޮ͍ͯ͘Δ͕ɼϩέʔγϣϯύ ϥϝʔλʹରͯ͠͸ෆม 26 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ظ଴஋ܭࢉͷྫ ਖ਼ن෼෍ͷظ଴஋ͱ෼ࢄ X ∼ N(µ, σ2) ͷͱ͖ɼE [X] , V [X] ΛٻΊΑɽ X ∼ N(µ, σ2) ΑΓɼ E [X] = ∫ R x 1 √ 2πσ exp [ − (x − µ)2 2σ2 ] dx = 1 √ 2π ∫ R (σz + µ) 1 σ exp [ − z2 2 ] σ dz ( z := x − µ σ ) = σ √ 2π ∫ R z exp [ − z2 2 ] dz + µ √ 2π ∫ R exp [ − z2 2 ] dz. ͜͜Ͱɼz exp(−z2/2) ͕حؔ਺Ͱ͋Δ͜ͱ͔Β ∫ R z exp [ − z2 2 ] = 0. 27 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ظ଴஋ܭࢉͷྫ ͞Βʹɼ ∫ R exp [ − z2 2 ] = √ 2π Ͱ͋Δ͔Βɼ E [X] = µ √ 2π · √ 2π = µ Ͱ͋Δɽ·ͨɼ V [X] = ∫ R (x − µ)2 1 √ 2πσ exp [ − (x − µ)2 2σ2 ] dx = ∫ R σ2z2 1 √ 2πσ exp [ − z2 2 ] σ dz ( z := x − µ σ ) = σ2 √ 2π ∫ R z2 exp [ − z2 2 ] dz = σ2 √ 2π ∫ R z ( exp [ − z2 2 ])′ dz = σ2 √ 2π [ z exp ( − z2 2 )]∞ ∞ + σ2 √ 2π ∫ R exp ( − z2 2 ) dz =σ2. 28 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ڞ෼ࢄ • ଟ࣍ݩͷಛੑ஋΋঺հ͓ͯ͘͠ ڞ෼ࢄ ֬཰ม਺ X, Y ʹର͠ɼE [X] < ∞, E [Y ] < ∞ ͱ͠ɼφ(x, y) = (x − E [X]) (y − E [Y ]) ͱ͢Δɽ͜ͷͱ͖ɼ Cov (X, Y ) := E [ϕ(X, Y )] = E [(X − E [X]) (Y − E [Y ])] Λ X ͱ Y ͷڞ෼ࢄͱ͍͏ɽ ڞ෼ࢄʹؔͯ͠ɼ Cov (X, Y ) = E [XY ] − E [X] E [Y ] , Cov (aX + b, cY + d) = acCov (X, Y ) , V [aX + bY ] = a2V [X] + b2V [Y ] + 2abCov (X, Y ) ͕੒Γཱͭɽ 29 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ૬ؔ܎਺ ૬ؔ܎਺ V [X] < ∞, V [Y ] < ∞ ͷͱ͖ɼ ρXY = Cov (X, Y ) √ V [X] √ V [Y ] Λ X ͱ Y ͷ૬ؔ܎਺ͱ͍͏ɽ • ρXY > 0 ͷͱ͖͸ʮਖ਼ͷ૬͕ؔ͋Δʯ ɼρXY < 0 ͷͱ͖͸ʮෛͷ૬ؔ ͕͋Δʯ ɼρXY = 0 ͷͱ͖͸ʮແ૬ؔͰ͋Δʯͱ͍͏ • Ұൠతʹ “ಠཱ =⇒ ແ૬ؔ” ͕੒Γཱͭ • [X, Y ] ͕ 2 ࣍ݩਖ਼ن෼෍ʹै͏ͱ͖͸ɼ“ແ૬ؔ =⇒ ಠཱ” ͕੒ཱ ͢Δ 30 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ 1 લճͷ෮श 2 ֬཰ม਺ͷಠཱੑ 3 ֬཰෼෍ͷಛੑ஋ 4 ੵ཰฼ؔ਺ 5 ۙࣅ๏ଇ 6 ࣍ճ༧ࠂ 31 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ϞʔϝϯτͱϞʔϝϯτ฼ؔ਺ • ֬཰ม਺ͷϞʔϝϯτ͸ɼ෼෍ͷܗΛ஌ΔͨΊͷई౓ͱͯ͠࢖ΘΕΔ • ྫ͑͹ɼඪ४Խͨ͠ޙͷݪ఺पΓͷ 3 ࣍Ϟʔϝϯτ͸࿪౓ͱ͍͍ɼ෼ ෍ͷ੄ͷॏ͞Λද͢ • ͱ͍͏͜ͱͰɼϞʔϝϯτ͸ॏཁ • ͔͠͠ɼ௚઀ p.m.f., p.d.f. ͔Βܭࢉ͢Δͷ͸େม 32 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ Ϟʔϝϯτ฼ؔ਺ Ϟʔϝϯτ฼ؔ਺ (moment generating function) ֬཰ม਺ X ͷϞʔϝϯτ฼ؔ਺ (m.g.f.) Λ gX (θ) = E [exp(θX)] Ͱఆٛ͢Δɽ • ద౰ͳ ε > 0 ΛऔΕ͹ɼ|θ| < ε ͳΔ θ ʹରͯ͠ E [exp(θX)] ͸༗ݶ ֬ఆͱ͍͏Ծఆ͕ඞཁ • ্هԾఆ͕ແ͍Α͏ͳɼΑΓҰൠతͳ΋ͷ (ಛੑؔ਺) ΋͋Δ • Ϟʔϝϯτ฼ؔ਺Λ θ Ͱ k ճඍ෼͢Δ͜ͱͰɼݪ఺पΓͷ k ࣍Ϟʔ ϝϯτΛٻΊΔ͜ͱ͕ग़དྷΔ 33 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ Ϟʔϝϯτ฼ؔ਺ m.g.f. ͷඍ෼ ֬཰ม਺ X ͷ m.g.f. Λ gX (θ) ͱ͠ɼθ ʹؔͯ͠ظ଴஋ه߸ԼͰ೚ ҙ࣍਺ͷඍ෼͕Մೳͱ͢Δɽ͜ͷͱ͖ɼ E [ Xk ] = dk dθk gX (0), k = 1, 2, . . . ͕੒Γཱͭɽ • m.g.f. ͷఆ͔ٛΒ͙͢ʹಋ͚Δɿ dk dθk gX (θ) = E [ ∂k ∂θk exp(θX) ] = E [ Xk exp(θX) ] ͱͳΔͷͰɼθ = 0 ͱ͢Ε͹໋୊ͷ͕ࣜ੒Γཱͭɽ□ 34 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ Ϟʔϝϯτ฼ؔ਺͔Βظ଴஋ɾ෼ࢄͷಋग़ ೋ߲෼෍ X ∼ B(n, p) ͷͱ͖ɼX ͷ m.g.f. Λ༻͍ͯظ଴஋ɾ෼ࢄΛٻΊΔɽ X ͷ m.g.f. ͸ gX (θ) = n ∑ x=0 exp(θx) ( n x ) px(1 − p)n−x = ( peθ + q )n , q := 1 − p Ͱ͋Δ͔Βɼ d dθ gX (θ) = npeθ ( peθ + q )n−1 , d2 d2θ gX (θ) = npeθ ( peθ + q ) + n(n − 1)p2 exp(2θ) ( peθ + q )n−1 Ͱ͋ΔɽΑͬͯɼ E [X] = d dθ gX (θ)(0) = np(p + q)n−1 = np, V [X] = d2 d2θ gX (0) = np(p + q) + n(n − 1)p2(p + q)n−1 = np + n(n − 1)p2. 35 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ 1 લճͷ෮श 2 ֬཰ม਺ͷಠཱੑ 3 ֬཰෼෍ͷಛੑ஋ 4 ੵ཰฼ؔ਺ 5 ۙࣅ๏ଇ 6 ࣍ճ༧ࠂ 36 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ౷ܭత઴ۙཧ࿦ • n ݸͷσʔλʹجͮ͘౷ܭղੳख๏ͷਫ਼౓Λਫ਼ີʹධՁ͢Δ͜ͱ͸ɼ n ͕খ͍͞ͱ೉͍͠ࣄ͕ଟ͍ • n ͕େ͖͍৔߹ͩͱɼn → ∞ ͷঢ়گͰಋ͔ΕΔ౷ܭత໋୊Λۙࣅͱ ͯ͠ར༻Ͱ͖Δ • ͜ͷΑ͏ͳ n → ∞ Ͱͷཧ࿦Λɼ౷ܭతۃݶཧ࿦ɾ౷ܭత઴ۙཧ࿦ͳ Ͳͱ͍͏ • ౷ܭత઴ۙཧ࿦ͷ౔୆ͱͳ͍ͬͯΔͷ͸֬཰࿦ʹ͓͚Δۃݶఆཧ 37 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ େ਺ͷऑ๏ଇ • ֬཰ม਺ X1 , . . . , Xn ͕ಠཱʹಉҰͷ෼෍ F ʹै͏ͱ͖ɼ X1 , . . . , Xn iid ∼ F 1 ͱॻ͘͜ͱʹ͢Δ • ্هͷ X1 , . . . , Xn ͸ແ࡞ҝඪຊͱ΋ݺ͹ΕΔ Weak Law of Large Numbers X1 , . . . , Xn , . . . Λฏۉ µɼ෼ࢄ σ2 Λ࣋ͭ෼෍͔Βͷແ࡞ҝඪຊͱ ͠ɼX = (1/n) ∑ n i=1 Xi ͱ͢Δɽ͜ͷͱ͖ɼ ∀ε > 0, lim n→∞ P ( X − µ > ε ) = 0 (1) ͕੒Γཱͭɽ 1independently and identically distributed ͷུ 38 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ େ਺ͷऑ๏ଇ –γϛϡϨʔγϣϯ– Figure: ֤ ε Ͱͷ P ( µ − X > ε ) Λඳ͍ͨɽn ͕େ͖͘ͳΔʹͭΕ µ − X > ε ͱͳΔ֬཰͕খ͘͞ͳ͍ͬͯΔ͜ͱ͕෼͔Δɽ 39 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ த৺ۃݶఆཧ Central Limit Theorem X1 , . . . , Xn , . . . Λฏۉ µɼ෼ࢄ σ2 Λ࣋ͭ෼෍͔Βͷແ࡞ҝඪຊͱ ͢Δɽ͜ͷͱ͖ɼ∀y ∈ R, lim n→∞ P ( X − µ σ/ √ n ≤ y ) = ∫ y −∞ 1 √ 2π exp ( − z2 2 ) dz (2) ͕੒Γཱͭɽ • ࣮ࡍʹ͸ X · ∼ N ( µ, σ2/n ) ͱ͍͏ۙࣅ͕༻͍ΒΕΔ͜ͱ͕ଟ͍ ( · ∼ ͸ۙࣅతʹै͏ͱ͍͏ҙຯ) 40 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ऩଋ ऩଋ Y, Y1 , Y2 , . . . Λ֬཰ม਺ྻɼY ∼ F, Fi ∼ Fi ͱ͢Δɽ ֬཰ऩଋ ∀ε > 0, lim n→∞ P (|Yn − Y | ≥ ε) = 0 ͕੒Γཱͭͱ͖ɼ Yn ͕ Y ʹ֬཰ऩଋ͢Δͱ͍͍ Yn P −→ Y ͱද͢ɽ ෼෍ऩଋ F ͷશͯͷ࿈ଓ఺ y Ͱ lim n→∞ Fn (y) = F(y) ͕੒Γཱͭͱ͖ɼYn ͕ Y ʹ෼෍ऩଋ͢Δͱ͍͍ɼYn D −→ Y ͱ ද͢ɽ 41 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ த৺ۃݶఆཧ –γϛϡϨʔγϣϯ– Step 1 ֬཰ม਺ x1 , . . . , x1000 ∼ χ2 k ͔Β x Λܭࢉ͢Δ Step 2 z = √ 1000(x − k)/(2k) Λܭࢉ͢Δ Step 3 Step 1, Step 2 Λ 5000 ճ܁Γฦ͠ɼz ͷ෼෍Λ֬ೝ͢Δ 42 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ த৺ۃݶఆཧ –γϛϡϨʔγϣϯ– Figure: த৺ۃݶఆཧ 43 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ऩଋ • WLLN ͸ X P −→ µ Λओு͍ͯ͠Δ • CLT ͸ √ n ( X − µ ) D −→ N (0, 1) Λओு͍ͯ͠Δ • WLLN, CLT ͱ΋ʹ i.i.d. Ͱ͋Ε͹੒ཱ͢Δ (ඇৗʹڧྗ) • ݩͷ෼෍͕ C-type Ͱ΋ D-type Ͱ΋ɼۃݶ෼෍͸ਖ਼ن෼෍ʹͳΔ • ਖ਼ن෼෍͕೗Կʹେࣄ͔Α͘෼͔ΔΑͶʁ 44 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ 1 લճͷ෮श 2 ֬཰ม਺ͷಠཱੑ 3 ֬཰෼෍ͷಛੑ஋ 4 ੵ཰฼ؔ਺ 5 ۙࣅ๏ଇ 6 ࣍ճ༧ࠂ 45 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ ࣍ճͷ಺༰ • ౷ܭతਪఆ 46 / 47

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લճͷ෮श ֬཰ม਺ͷಠཱੑ ֬཰෼෍ͷಛੑ஋ ੵ཰฼ؔ਺ ۙࣅ๏ଇ ࣍ճ༧ࠂ Reference • ੺ฏণจ, ౷ܭղੳೖ໳, ৿๺ग़൛גࣜձࣾ, 2007 • தాण෉, ಺౻؏ଠ, ֬཰ɾ౷ܭ, ֶज़ਤॻग़൛ࣾ, 2016 • ಺౻؏ଠ, ਺ཧ౷ܭֶ II ߨٛϊʔτ, 2014 • S.E. γϡϦʔϰ, ϑΝΠφϯεͷͨΊͷ֬཰ղੳ II ࿈ଓ࣌ؒϞσϧ, 2014 47 / 47