266 Appendice C - Covarianza e correlazione
lineari delle stesse variabili casuali, sono tra loro correlate (come osservato nel paragrafo C.1 ); le equazioni (11.10) possono essere riscritte (l’abbiamo già visto nel paragrafo 11.4.1 ) come
a
=
∑
i
a
i
y
i
{\displaystyle a=\sum \nolimits _{i}a_{i}\,y_{i}}
e
b
=
∑
i
b
i
y
i
{\displaystyle b=\sum \nolimits _{i}b_{i}\,y_{i}}
una volta che si sia posto
{
a
i
=
1
Δ
[
∑
j
x
j
2
−
(
∑
j
x
j
)
x
i
]
b
i
=
1
Δ
[
N
x
i
−
∑
j
x
j
]
{\displaystyle {\begin{cases}a_{i}=\displaystyle {\frac {1}{\Delta }}\,\left[\sum \nolimits _{j}{x_{j}}^{2}-\left(\sum \nolimits _{j}x_{j}\right)x_{i}\right]\\b_{i}=\displaystyle {\frac {1}{\Delta }}\left[N\,x_{i}-\sum \nolimits _{j}x_{j}\right]\end{cases}}}
Ricordando la definizione di
Δ
{\displaystyle \Delta }
Δ
{\displaystyle \Delta }
=
N
(
∑
j
x
j
2
)
−
(
∑
j
x
j
)
2
{\displaystyle =N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)-\left(\sum \nolimits _{j}x_{j}\right)^{2}}
=
N
2
[
∑
j
x
j
2
N
−
(
∑
j
x
j
N
)
2
]
{\displaystyle =N^{2}\left[{\frac {\sum \nolimits _{j}{x_{j}}^{2}}{N}}-\left({\frac {\sum \nolimits _{j}x_{j}}{N}}\right)^{2}\right]}
ed infine come
Visto che le
y
i
{\displaystyle y_{i}}
(C.4) ; ne ricaviamo, sostituendovi la (C.10) e ricordando che gli errori sono tutti uguali, che
Cov
(
a
,
b
)
{\displaystyle {\text{Cov}}(a,b)}
=
∑
i
a
i
b
i
Var
(
y
i
)
{\displaystyle =\sum \nolimits _{i}a_{i}\,b_{i}\,{\text{Var}}(y_{i})}
=
σ
y
2
Δ
2
∑
i
[
∑
j
x
j
2
−
(
∑
j
x
j
)
x
i
]
[
N
x
i
−
∑
j
x
j
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\,\sum \nolimits _{i}\left[\sum \nolimits _{j}{x_{j}}^{2}-\left(\sum \nolimits _{j}x_{j}\right)x_{i}\right]\left[Nx_{i}-\sum \nolimits _{j}x_{j}\right]}
=
σ
y
2
Δ
2
∑
i
[
(
∑
j
x
j
2
)
N
x
i
−
(
∑
j
x
j
2
)
(
∑
j
x
j
)
−
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\,\sum \nolimits _{i}{\Biggl [}\left(\sum \nolimits _{j}{x_{j}}^{2}\right)Nx_{i}-\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)-}
−
(
∑
j
x
j
)
N
x
i
2
+
(
∑
j
x
j
)
2
x
i
]
{\displaystyle -\left(\sum \nolimits _{j}x_{j}\right)N{x_{i}}^{2}+\left(\sum \nolimits _{j}x_{j}\right)^{2}\!x_{i}{\Biggr ]}}
=
σ
y
2
Δ
2
[
N
(
∑
j
x
j
2
)
(
∑
i
x
i
)
−
N
(
∑
j
x
j
2
)
(
∑
j
x
j
)
−
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\,{\Biggl [}N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left(\sum \nolimits _{i}x_{i}\right)-N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)-}
−
N
(
∑
j
x
j
)
(
∑
i
x
i
2
)
+
(
∑
j
x
j
)
3
]
{\displaystyle -N\left(\sum \nolimits _{j}x_{j}\right)\left(\sum \nolimits _{i}{x_{i}}^{2}\right)+\left(\sum \nolimits _{j}x_{j}\right)^{3}{\Biggr ]}}
=
σ
y
2
Δ
2
(
∑
j
x
j
)
[
(
∑
j
x
j
)
2
−
N
(
∑
j
x
j
2
)
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\,\left(\sum \nolimits _{j}x_{j}\right)\,\left[\left(\sum \nolimits _{j}x_{j}\right)^{2}-N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\right]}
![{\displaystyle {\begin{cases}a_{i}=\displaystyle {\frac {1}{\Delta }}\,\left[\sum \nolimits _{j}{x_{j}}^{2}-\left(\sum \nolimits _{j}x_{j}\right)x_{i}\right]\\b_{i}=\displaystyle {\frac {1}{\Delta }}\left[N\,x_{i}-\sum \nolimits _{j}x_{j}\right]\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fda5c61330dae6fe1e5978c8b73ba89193be240)
![{\displaystyle =N^{2}\left[{\frac {\sum \nolimits _{j}{x_{j}}^{2}}{N}}-\left({\frac {\sum \nolimits _{j}x_{j}}{N}}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5e33fb1f23b7ccf8049c0ce6cd8c6486f9797e)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\,\sum \nolimits _{i}\left[\sum \nolimits _{j}{x_{j}}^{2}-\left(\sum \nolimits _{j}x_{j}\right)x_{i}\right]\left[Nx_{i}-\sum \nolimits _{j}x_{j}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd3e1006ccd4c9d408a592390821d0b51fd0b47)
![{\displaystyle -\left(\sum \nolimits _{j}x_{j}\right)N{x_{i}}^{2}+\left(\sum \nolimits _{j}x_{j}\right)^{2}\!x_{i}{\Biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c42e1789113f82f71f3b014a9ff466b480f1ea2)
![{\displaystyle -N\left(\sum \nolimits _{j}x_{j}\right)\left(\sum \nolimits _{i}{x_{i}}^{2}\right)+\left(\sum \nolimits _{j}x_{j}\right)^{3}{\Biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2289051a24ab3f99ee0d82d87df75fb5358491d6)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\,\left(\sum \nolimits _{j}x_{j}\right)\,\left[\left(\sum \nolimits _{j}x_{j}\right)^{2}-N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/554dc72d7352936324796e8bed5613112795199b)
