, xn } Oݸͷσʔλ FHOਓ͍Δͱ͋ΔΫϥεͷςετͷͳͲ جຊ౷ܭྔ ¯ x = 1 n n X i=1 xi ฏۉ 2 = 1 n n X i=1 ( xi ¯ x )2 ࢄ = v u u t 1 n n X i=1 ( xi ¯ x )2 ඪ४ภࠩ ˞ ਪଌΛߦ͏࣌ͷࢄ O ͰׂΔͱ͍͏ ͕͋Γ·͕ͦ͢Εޙ΄ͲͷϖʔδͰɻ
= n ! ( n x )! x ! E[ X ] = n X x =0 xP ( x ) = n X x =0 x ✓ n x ◆ px(1 p )n x = n X x =0 x n ! ( n x )! x !px(1 p )n x = n X x =0 n ( n 1)! ( n x )!( x 1)!px(1 p )n x = np n X x =0 ✓ n 1 m 1 ◆ p ( x 1)(1 p )( n 1) ( x 1) = np
Var[ cX ] = Z 1 1 ( cx E[ cx ])2 f ( x ) dx = Z 1 1 ( cx cµ )2 f ( x ) dx = Z 1 1 c 2( x µ )2 f ( x ) dx = c 2 Z 1 1 ( x µ )2 f ( x ) dx = c 2Var[ X ] ޙͰ͏ܭࢉ
cnxn] = E[{( c1x1 + · · · + cnxn) E[ c1x1 + · · · + cnxn]}2] = E[{ c1( x1 µ1) + · · · + c1( x1 µ1)}2] = E[ n X i=1 c 2 i ( xi µi)2 + X i6=j cicj( xi µj)( xi µj)] = n X i=1 c 2 i E[( xi µi)2] + X i6=j cicjE[( xi µj)( xi µj)] = c 2 1 2 1 + · · · + c 2 n 2 n c1µ1 + · · · + cnµn = E[ xi µi]E[ xj µj] = 0 ಠཱͳͷͰ = 2 i
)2 f ( x ) dx = Z I1 ( x µ )2 f ( x ) dx + Z I2 ( x µ )2 f ( x ) dx + Z I3 ( x µ )2 f ( x ) dx 㱢ͳͷͰ͜ΕΛফͯ͠ෆࣜʹ͢Δɻ 2 = Z I1 ( x µ )2 f ( x ) dx + Z I3 ( x µ )2 f ( x ) dx = Z I1 2 2 f ( x ) dx + Z I3 2 2 f ( x ) dx = 2 2[ P ( x 2 I1) + P ( x 2 I3)] ͱ͢Δͱɺ I1 = ( 1, µ ), I2 = [µ , µ + ], I3 = (µ + , 1) = P (| x µ | > ) P (| x µ | > ) 5 1 2 )
E[ ext] = Z 1 1 ext f ( x ) dx ΛUͷؔͱͯ͠Έͨ࣌ɺ͜ΕΛ֬มYͷੵؔ .PNFOU(FOFSBUJOH'VODUJPO ͱ͍͏ɻ M x (t) = E[ext] ੵؔͷॏཁͳੑ࣭ ֬มͷੵؔͱ֬มͷੵؔ ͕ͷۙͰҰக͢Δͱ͖ɺͦͷ̎ͭͷ֬ มಉ֬͡ʹै͏ɻ ূ໌ུ M x (t) My(t) x t = 0 y ˞ੵ͕ؔଘࡏ͢ΔͨΊʹͯ͢ͷ࣍Ͱੵ͕ଘࡏ͢Δඞཁ͕͋Δɻ
· · · , xn) µ 2 ¯ x µ / p n , n ! 1 ඪ४ਖ਼نɹɹɹɹͰ͋Δɻ N(0, 1) T = x1 + · · · + xn T nµ p n ࠓɹɹɹɹɹɹ͕ฏۉ̌ɺࢄɹͷ ਖ਼نʹै͏͜ͱΛূ໌͠·͢ɻ 2 T 0 = T nµ p n = ¯ x µ 1 / p n
nµ p n + · · · + xn µ p n ͷੵؔΛߟ͑Δɻ ࣍ʹ = n X i=1 xi µ p n MT 0 (t) = MP n i =1 ⇣ xi µ p n ⌘(t) = E[e P n i =1 ⇣ xi µ p n ⌘ t ] = n Y i=0 E[e ⇣ xi µ p n ⌘ t ] = ✓ 1 + 1 n 2t2 + n 2 ◆n ಠཱͳͷͰ FͷఆٛΑΓ er ⌘ lim n!1 ⇣ 1 + r n ⌘n r r = lim n!1 ⇣ 1 + r n ⌘n
= n · 1 n µ = µ ʮඪຊநग़ʯʹͯطʹݟͨΑ͏ʹظΛͱΔͱ ¯ x ͷΑ͏ʹɺύϥϝʔλʔͱ͘͠ͳΔͨΊɺฏۉʹؔ͢Δ ෆภਪఆྔͰ͋Δɻ ࣍ϖʔδͰࣔ͢Α͏ʹɺࢄͷෆภਪఆྔOͰׂΔͷͰͳ͘ OͰׂΔ s 2 = 1 n 1 n X i=1 ( xi ¯ x )2 ͱͳΔɻ ຊεϥΠυͷ࠷ॳͷ෦Ͱɺਪଌͷ߹OͰͳ͘OͰׂΔ ͱॻ͍ͨཧ༝͜ͷෆภੑΛ࣋ͨͤΔͨΊͰ͋ͬͨɻ
· , xn) xi f ( xi) n Y i=1 f ( xi) n Y i=1 f ( xi | ✓ ) Ͱ͋ΔɻύϥϝʔλʔВΛ໌ࣔ͢Δͱ ͱͳΔɻ͜ͷ࣌ɺؔͷܗશ͘ಉ͡ͰɺมΛ Вͱ͠ɺɹΛݻఆͱͯ͠Έͨͱ͖ɺ͜ΕΛؔͱ͍͍ɺ ԼهͷΑ͏ʹද͢ɻ xi ` ( ✓ | x1, x2, · · · , xn) = n Y i=1 f ( xi | ✓ )
x2, · · · , x10) = n 2 (2⇡) n 2 log 2 1 2 2 n X i=1 (xi µ) 2 @L @µ = 1 2 2 n X i=1 ( xi µ )2 ) n X i=1 xi = nµ ) µ ⇤ = 1 n n X i=1 xi `(µ, 2| x1, x2, · · · , xn) = n Y i=1 1 p 2⇡ 2 exp ✓ 1 2 (xi µ) 2 2 ◆
1) v ⇠ 2(m) v > 0 1 < u < +1 f ( u, v ) = 1 p 2 ⇡ exp ✓ u2 2 ◆ (1 / 2) n/2 ( n/ 2) vn/2 1e v/2 ͱͳΔɻ͜͜Ͱมม ΧΠೋͷີؔ ඪ४ਖ਼نͷີؔ t = u p v/m x = v ɺɹɹɹɹΛ͓͜ͳ͍ࣜมܗͷͷͪUʹؔ͢Δ f(t) = m+1 2 p m⇡ m 2 ✓ t2 m + 1 ◆ m+1 2 ɹ͕ಘΒΕΔɻ ˞͖ͪΜͱͨ͠ূ໌ʮֶγϦʔζཧ౷ܭֶ Ҵ֞એੜ ʯɹQςΟʔࢀর ΨϯϚؔ (z) = Z 1 0 tz 1e t dt
· · · , xn) xi ⇠ N ( µ, 2) ¯ x ⇠ N ( µ, 2 /n ) ¯ x ˞ূ໌ʮֶγϦʔζཧ౷ܭֶ Ҵ֞એੜ ʯɹQඪຊฏۉͱඪຊࢄͷࢀর 1 2 n X i=1 ( xi ¯ x )2 ⇠ 2 n 1 ·ͨɺඪຊฏۉͱͷภࠩͷೋΛࢄͰׂͬͨͷɺ ࣗ༝OͷΧΠೋʹै͏ɻ
n ⇠ N (0 , 1) v = 1 2 n X i=1 ( xi ¯ x )2 ⇠ 2 n 1 ͱ͢Δͱɺ t = u p v/ ( n 1) = ¯ x µ / p n · " 1 2 1 ( n 1) n X i=1 ( xi ¯ x )2 # 1/2 = ¯ x µ 1 / p n · 1 p s 2 = ¯ x µ s/ p n ⇠ tn 1 s 2 = 1 n 1 n X i=1 ( xi ¯ x )2 ෆภࢄ s2 ˢࢄМ͕ফ͑ͨʂ
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