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}}
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.
![{\displaystyle \mathbb {E} [R_{P}]=\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f892f670bf130a4a19c1a0cdb28f929af85e3ab)
![{\displaystyle \underbrace {{\text{Var}}(\mathbb {E} [R_{P}])} _{\equiv \sigma _{P}^{2}}=\mathbb {E} [R_{P}-\mathbb {E} [R_{P}]]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d54fe3133cf94bc18b0bd3437aef5e7c78fefa0)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bda6ad1b21edb646dd894cdfe6979d0d6c8eef76)
![{\displaystyle \sigma _{P}^{2}=\mathbb {E} [\sum _{i=1}^{n}x_{i}(R_{i}-\mathbb {E} [R_{i}])]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53c229a60d42c776c1927263e934f73575631fea)
![{\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}])]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/803b31452751cd604df1d58e5fb7aaf3a4099bc4)
![{\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}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b31d9252f35e7db4585ba732938a079d4ae5e7b3)
