Laghdú rioscaí trí éagsúlú
cuir in eagar
Seo an aisíoc ionchais ar phunann:
E [ R P ] = ∑ i = 1 n x i E [ R i ] {\displaystyle \mathbb {E} [R_{P}]=\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]} (tá x i {\displaystyle x_{i}} an chomhréir rachmais san urrús i {\displaystyle i} ).
Seo an athraitheas punainne:
Var ( E [ R P ] ) ⏟ ≡ σ P 2 = E [ R P − E [ R P ] ] 2 {\displaystyle \underbrace {{\text{Var}}(\mathbb {E} [R_{P}])} _{\equiv \sigma _{P}^{2}}=\mathbb {E} [R_{P}-\mathbb {E} [R_{P}]]^{2}}
σ P 2 = E [ ∑ i = 1 n x i R i − E [ ∑ i = 1 n x i E [ R i ] ] ] 2 {\displaystyle \sigma _{P}^{2}=\mathbb {E} [\sum _{i=1}^{n}x_{i}R_{i}-\mathbb {E} [\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]]]^{2}} σ P 2 = E [ ∑ i = 1 n x i ( R i − E [ R i ] ) ] 2 {\displaystyle \sigma _{P}^{2}=\mathbb {E} [\sum _{i=1}^{n}x_{i}(R_{i}-\mathbb {E} [R_{i}])]^{2}} σ P 2 = E [ ∑ i = 1 n ∑ j = 1 n x i x j ( R i − E [ R i ] ) ( R j − E [ R j ] ) ] {\displaystyle \sigma _{P}^{2}=\mathbb {E} [\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])]} σ P 2 = E [ ∑ i = 1 n x i 2 ( R i − E [ R i ] ) 2 ⏟ ≡ σ i 2 + ∑ i = 1 n ∑ j = 1 , i ≠ j n x i x j ( R i − E [ R i ] ) ( R j − E [ R j ] ) ⏟ ≡ σ i j ] {\displaystyle \sigma _{P}^{2}=\mathbb {E} [\sum _{i=1}^{n}x_{i}^{2}\underbrace {(R_{i}-\mathbb {E} [R_{i}])^{2}} _{\equiv \sigma _{i}^{2}}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\underbrace {(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])} _{\equiv \sigma _{ij}}]} σ P 2 = ∑ i = 1 n x i 2 σ i 2 + ∑ i = 1 n ∑ j = 1 , i ≠ j n x i x j σ i j {\displaystyle \sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\sigma _{i}^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\sigma _{ij}} I bpunann chomhualaithe, x i = x j = 1 n , ∀ i , j {\displaystyle x_{i}=x_{j}={\frac {1}{n}},\forall i,j} .
σ P 2 = n 1 n 2 σ i 2 + n ( n − 1 ) 1 n 1 n σ i j {\displaystyle \sigma _{P}^{2}=n{\frac {1}{n^{2}}}\sigma _{i}^{2}+n(n-1){\frac {1}{n}}{\frac {1}{n}}\sigma _{ij}} σ P 2 = 1 n σ i 2 + n − 1 n σ i j {\displaystyle \sigma _{P}^{2}={\frac {1}{n}}\sigma _{i}^{2}+{\frac {n-1}{n}}\sigma _{ij}} lim n → ∞ σ P 2 = σ ¯ i j {\displaystyle \lim _{n\rightarrow \infty }\sigma _{P}^{2}={\bar {\sigma }}_{ij}} Dá bhrí sin, nuair a théann an uimhir urrúis go dtí éigríoch, druidinn an athraitheas punainne go dtí an meánchomhathraitheas idir na hurrúis - seo é riosca córasach.