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多変量正規分布に従う確率変数の条件付き期待値・分散

75eb40de505a993c792ff3bc6683dc0b?s=47 axjack
January 11, 2022

 多変量正規分布に従う確率変数の条件付き期待値・分散

多変量正規分布に従う確率変数の条件付き期待値・分散

75eb40de505a993c792ff3bc6683dc0b?s=128

axjack

January 11, 2022
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  1. ଟมྔਖ਼ن෼෍ʹै͏֬཰ม਺ͷ ৚݅෇͖ظ଴஋ɾ෼ࢄ  4BUPBLJ/PHVDIJ BYKBDL!HNBJMDPN  1

  2. ͱ͠ɺ9͸ฏۉЖɾ෼ࢄڞ෼ࢄߦྻЄ ͷଟมྔਖ਼ن෼෍ ʹै͏ͱ͢Δɻ ͜͜Ͱɺ ɹɾ9Λ̎෼ׂ ɹɾЖΛ̎෼ׂ ɹɾЄΛ̐෼ׂ ͓ͯ͘͠ɻ ४උ Λ֬཰ม਺ϕΫτϧ

    Λظ଴஋ϕΫτϧ Λ෼ࢄڞ෼ࢄߦྻ Σ = ( Σ11 Σ12 Σ21 Σ22 ) X μ Σ X = ( X1 X2 ) μ = ( μ1 μ2 ) X ∼ N(μ, Σ) μi = E[Xi ] ͨͩ͠ Σij = Cov[Xi , Xj ] ͨͩ͠  2
  3. ެࣜ ৚݅෇͖֬཰ม਺ͷ ظ଴஋ɾ෼ࢄ E[X1 |X2 = x2 ] = μ1

    + Σ12 Σ22 −1(x2 − μ2 ) V[X1 |X2 = x2 ] = Σ11 − Σ12 Σ22 −1Σ21 X1 |X2 = x2 Λɺ9YͰ৚͚݅ͮͨ9ͷ֬཰ม਺ͱ͢Δɻ ͜ͷ࣌ɺ9c9Yͷظ଴஋ɾ෼ࢄ͸ҎԼͰ͋Δɻ ˞ࢀߟɿʰ೔ຊ౷ܭֶձެࣜೝఆɹ౷ܭݕఆ̍ڃରԠɹ౷ܭֶʱ೔ຊ౷ܭֶձɹฤ Qఆཧ  3
  4. ྫ୊ ( X Y Z ) ∼ N (( 1

    2 3 ) , ( 2 0 1 0 3 2 1 2 4 )) ( X Y Z ) ̏มྔ֬཰ม਺ ͸̏มྔਖ਼ن෼෍ ʹै͏ͱ͢Δɻ ͜ͷ࣌ɺ Z|X = x, Y = y X, Y|Z = z   ʹ͓͚Δɺظ଴஋ɾ෼ࢄΛٻΊΑɻ ˞ࢀߟ౷ܭݕఆ४̍ڃ೥݄໰  4
  5.  ͷղ౴ μ = ( 3 1 2 ) Σ

    = ( 4 1 2 1 2 0 2 0 3 ) μ1 = E[Z] μ2 = E[(X Y)′  ] Σ11 Σ12 Σ22 Σ21 ( X1 X2 ) ∼ N (( μ1 μ2 ), ( Σ11 Σ12 Σ21 Σ22 )) E[X1 |X2 = x2 ] = μ1 + Σ12 Σ22 −1(x2 − μ2 ) V[X1 |X2 = x2 ] = Σ11 − Σ12 Σ22 −1Σ21 ( Z X Y ) ∼ N (( 3 1 2 ) , ( 4 1 2 1 2 0 2 0 3 )) ΑΓɺ E[Z|(X = x, Y = y)] = μ1 + Σ12 Σ22 −1 ( x − 1 y − 2) = 3 + (1 2) ( 2 0 0 3) −1 ( x − 1 y − 2) V[Z |(X = x, Y = y)] = Σ11 − Σ12 Σ22 −1Σ21 = 4 − (1 2) ( 2 0 0 3) −1 ( 1 2)  5
  6.  ͷղ౴ μ = ( 1 2 3 ) Σ

    = ( 2 0 1 0 3 2 1 2 4 ) μ1 = E[(X Y)′  ] μ2 = E[Z] Σ11 Σ12 Σ22 Σ21 ( X1 X2 ) ∼ N (( μ1 μ2 ), ( Σ11 Σ12 Σ21 Σ22 )) E[X1 |X2 = x2 ] = μ1 + Σ12 Σ22 −1(x2 − μ2 ) V[X1 |X2 = x2 ] = Σ11 − Σ12 Σ22 −1Σ21 ( X Y Z ) ∼ N (( 1 2 3 ) , ( 2 0 1 0 3 2 1 2 4 )) ΑΓɺ E[X, Y |Z = z] = ( 1 2) + ( 1 2) 4−1 (z − 3) V[X, Y |Z = z] = Σ11 − Σ12 Σ22 −1Σ21 = ( 2 0 0 3) − ( 1 2) 4−1 (1 2)  6