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Teoria degli errori e fondamenti di statistica.djvu/198
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182
Capitolo 11 - Stime di parametri
σ
a
2
{\displaystyle {\sigma _{a}}^{2}}
=
∑
i
a
i
2
σ
y
2
{\displaystyle =\sum \nolimits _{i}{a_{i}}^{2}\,{\sigma _{y}}^{2}}
=
σ
y
2
∑
i
[
1
Δ
(
∑
j
x
j
2
−
x
i
∑
j
x
j
)
]
2
{\displaystyle ={\sigma _{y}}^{2}\;\sum \nolimits _{i}\left[{\frac {1}{\Delta }}\left(\sum \nolimits _{j}{x_{j}}^{2}-x_{i}\sum \nolimits _{j}x_{j}\right)\right]^{2}}
=
σ
y
2
Δ
2
∑
i
[
(
∑
j
x
j
2
)
2
+
x
i
2
(
∑
j
x
j
)
2
−
2
x
i
(
∑
j
x
j
2
)
(
∑
j
x
j
)
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\;\sum \nolimits _{i}\left[\left(\sum \nolimits _{j}{x_{j}}^{2}\right)^{2}+{x_{i}}^{2}\left(\sum \nolimits _{j}x_{j}\right)^{2}-2\,x_{i}\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)\right]}
=
σ
y
2
Δ
2
[
N
(
∑
j
x
j
2
)
2
+
(
∑
i
x
i
2
)
(
∑
j
x
j
)
2
−
2
(
∑
j
x
j
2
)
(
∑
j
x
j
)
2
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)^{2}\!+\left(\sum \nolimits _{i}{x_{i}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)^{2}\!-2\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}
=
σ
y
2
Δ
2
[
N
(
∑
j
x
j
2
)
2
−
(
∑
j
x
j
)
2
(
∑
j
x
j
2
)
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)^{2}-\left(\sum \nolimits _{j}x_{j}\right)^{2}\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\right]}
=
σ
y
2
Δ
2
(
∑
j
x
j
2
)
[
N
(
∑
j
x
j
2
)
−
(
∑
j
x
j
)
2
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left[N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)-\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}
=
σ
y
2
∑
j
x
j
2
Δ
{\displaystyle ={\sigma _{y}}^{2}\;{\frac {\sum _{j}{x_{j}}^{2}}{\Delta }}}
e, similmente, per
b
{\displaystyle b}
:
σ
b
2
{\displaystyle {\sigma _{b}}^{2}}
=
∑
i
b
i
2
σ
y
2
{\displaystyle =\sum \nolimits _{i}{b_{i}}^{2}\,{\sigma _{y}}^{2}}
=
σ
y
2
∑
i
[
1
Δ
(
N
x
i
−
∑
j
x
j
)
]
2
{\displaystyle ={\sigma _{y}}^{2}\;\sum \nolimits _{i}\left[{\frac {1}{\Delta }}\left(N\,x_{i}-\sum \nolimits _{j}x_{j}\right)\right]^{2}}
=
σ
y
2
Δ
2
∑
i
[
N
2
x
i
2
+
(
∑
j
x
j
)
2
−
2
N
x
i
∑
j
x
j
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\;\sum \nolimits _{i}\left[N^{2}{x_{i}}^{2}+\left(\sum \nolimits _{j}x_{j}\right)^{2}-2\,N\,x_{i}\sum \nolimits _{j}x_{j}\right]}
=
σ
y
2
Δ
2
[
N
2
(
∑
i
x
i
2
)
+
N
(
∑
j
x
j
)
2
−
2
N
(
∑
j
x
j
)
2
]
{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N^{2}\left(\sum \nolimits _{i}{x_{i}}^{2}\right)+N\left(\sum \nolimits _{j}x_{j}\right)^{2}-2\,N\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}
=
N
σ
y
2
Δ
2
[
N
(
∑
i
x
i
2
)
−
(
∑
j
x
j
)
2
]
{\displaystyle ={\frac {N\,{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N\left(\sum \nolimits _{i}{x_{i}}^{2}\right)-\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}
=
σ
y
2
N
Δ
{\displaystyle ={\sigma _{y}}^{2}\;{\frac {N}{\Delta }}}
.
![{\displaystyle ={\sigma _{y}}^{2}\;\sum \nolimits _{i}\left[{\frac {1}{\Delta }}\left(\sum \nolimits _{j}{x_{j}}^{2}-x_{i}\sum \nolimits _{j}x_{j}\right)\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1f25495eeea0d3b49f6f6a054ae16d0c3501a2)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\;\sum \nolimits _{i}\left[\left(\sum \nolimits _{j}{x_{j}}^{2}\right)^{2}+{x_{i}}^{2}\left(\sum \nolimits _{j}x_{j}\right)^{2}-2\,x_{i}\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db48869cd0465dd96a600806f4bae0369dcb4ce7)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)^{2}\!+\left(\sum \nolimits _{i}{x_{i}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)^{2}\!-2\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc3420e44b47036ae4e7d00897f062181089b13)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)^{2}-\left(\sum \nolimits _{j}x_{j}\right)^{2}\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d001319afdbfc2112bbf3e2377f444418af76b6)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left(\sum \nolimits _{j}{x_{j}}^{2}\right)\left[N\left(\sum \nolimits _{j}{x_{j}}^{2}\right)-\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f727a01117790e62cfe92c806c64177666e91fac)
:
![{\displaystyle ={\sigma _{y}}^{2}\;\sum \nolimits _{i}\left[{\frac {1}{\Delta }}\left(N\,x_{i}-\sum \nolimits _{j}x_{j}\right)\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ccae59801a92821d17dd332affb1d51dfea6ff1)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\;\sum \nolimits _{i}\left[N^{2}{x_{i}}^{2}+\left(\sum \nolimits _{j}x_{j}\right)^{2}-2\,N\,x_{i}\sum \nolimits _{j}x_{j}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9885df50dac1d974b0d26f52b9510012c6016ad1)
![{\displaystyle ={\frac {{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N^{2}\left(\sum \nolimits _{i}{x_{i}}^{2}\right)+N\left(\sum \nolimits _{j}x_{j}\right)^{2}-2\,N\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd96981de06c5ce531cba62440556835697c1bb)
![{\displaystyle ={\frac {N\,{\sigma _{y}}^{2}}{\Delta ^{2}}}\left[N\left(\sum \nolimits _{i}{x_{i}}^{2}\right)-\left(\sum \nolimits _{j}x_{j}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0fb646f65b4814840090e328ac7674285418bf8)
.
