252 Appendice B - L'errore della varianza
in numero di
N
{\displaystyle N}
N
(
N
−
1
)
/
2
{\displaystyle N\,(N-1)/2}
C
2
4
{\displaystyle C_{2}^{4}}
i
{\displaystyle i}
j
{\displaystyle j}
k
{\displaystyle k}
l
{\displaystyle l}
E
(
∑
i
x
i
)
4
=
N
E
(
x
4
)
+
3
N
(
N
−
1
)
[
E
(
x
2
)
]
2
{\displaystyle E\left(\sum \nolimits _{i}x_{i}\right)^{4}=N\,E{\bigl (}x^{4}{\bigr )}+3\,N\,(N-1)\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}}
e, riprendendo la formule di partenza,
E
(
s
4
)
=
(
N
−
1
)
2
N
3
E
(
x
4
)
+
(
N
−
1
)
(
N
2
−
2
N
+
3
)
N
3
[
E
(
x
2
)
]
2
.
{\displaystyle E{\bigl (}s^{4}{\bigr )}={\frac {(N-1)^{2}}{N^{3}}}\,E{\bigl (}x^{4}{\bigr )}+{\frac {(N-1)(N^{2}-2N+3)}{N^{3}}}\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}.}
Per il valore medio di
s
2
{\displaystyle s^{2}}
E
(
s
2
)
=
σ
2
−
σ
x
¯
2
{\displaystyle E{\bigl (}s^{2}{\bigr )}=\sigma ^{2}-{\sigma _{\bar {x}}}^{2}}
inoltre
σ
2
=
E
{
(
x
−
x
∗
)
2
}
=
E
(
x
2
)
{\displaystyle \sigma ^{2}\;=\;E\left\{\left(x-x^{*}\right)^{2}\right\}\;=\;E{\bigl (}x^{2}{\bigr )}}
(essendo
x
∗
=
0
{\displaystyle x^{*}=0}
σ
x
¯
2
=
E
{
(
x
¯
−
x
∗
)
2
}
=
σ
2
N
{\displaystyle {\sigma _{\bar {x}}}^{2}\;=\;E\left\{\left({\bar {x}}-x^{*}\right)^{2}\right\}\;=\;{\frac {\sigma ^{2}}{N}}}
da cui abbiamo ottenuto a suo tempo la
E
(
s
2
)
=
N
−
1
N
σ
2
=
N
−
1
N
E
(
x
2
)
{\displaystyle E{\bigl (}s^{2}{\bigr )}\;=\;{\frac {N-1}{N}}\,\sigma ^{2}\;=\;{\frac {N-1}{N}}\,E{\bigl (}x^{2}{\bigr )}}
Per la varianza di
s
2
{\displaystyle s^{2}}
Var
(
s
2
)
{\displaystyle {\text{Var}}{\bigl (}s^{2}{\bigr )}}
=
E
(
s
4
)
−
[
E
(
s
2
)
]
2
{\displaystyle =E{\bigl (}s^{4}{\bigr )}-\left[E{\bigl (}s^{2}{\bigr )}\right]^{2}}
=
(
N
−
1
)
2
N
3
E
(
x
4
)
+
{\displaystyle ={\frac {(N-1)^{2}}{N^{3}}}\,E{\bigl (}x^{4}{\bigr )}+}
+
[
(
N
−
1
)
(
N
2
−
2
N
+
3
)
N
3
−
(
N
−
1
)
2
N
2
]
[
E
(
x
2
)
]
2
{\displaystyle +\left[\,{\frac {(N-1)(N^{2}-2N+3)}{N^{3}}}\,-\,{\frac {(N-1)^{2}}{N^{2}}}\,\right]\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}}
=
(
N
−
1
)
2
N
3
E
(
x
4
)
−
(
N
−
1
)
(
N
−
3
)
N
3
[
E
(
x
2
)
]
2
{\displaystyle ={\frac {(N-1)^{2}}{N^{3}}}\,E{\bigl (}x^{4}{\bigr )}\,-\,{\frac {(N-1)(N-3)}{N^{3}}}\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}}
=
N
−
1
N
3
{
(
N
−
1
)
E
(
x
4
)
−
(
N
−
3
)
[
E
(
x
2
)
]
2
}
{\displaystyle ={\frac {N-1}{N^{3}}}\,\left\{(N-1)\,E{\bigl (}x^{4}{\bigr )}\,-\,(N-3)\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}\right\}}
![{\displaystyle E\left(\sum \nolimits _{i}x_{i}\right)^{4}=N\,E{\bigl (}x^{4}{\bigr )}+3\,N\,(N-1)\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/719c9633c1925268d1749d89e76736446fd0c00c)
![{\displaystyle E{\bigl (}s^{4}{\bigr )}={\frac {(N-1)^{2}}{N^{3}}}\,E{\bigl (}x^{4}{\bigr )}+{\frac {(N-1)(N^{2}-2N+3)}{N^{3}}}\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20ac34c147dd9498cc88ca796fb1a81448718c86)
![{\displaystyle =E{\bigl (}s^{4}{\bigr )}-\left[E{\bigl (}s^{2}{\bigr )}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/482151270e3da50525adb369eeb0b08a72c9526b)
![{\displaystyle +\left[\,{\frac {(N-1)(N^{2}-2N+3)}{N^{3}}}\,-\,{\frac {(N-1)^{2}}{N^{2}}}\,\right]\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3660b2e1b8482464d49d60856c33de7aea849478)
![{\displaystyle ={\frac {(N-1)^{2}}{N^{3}}}\,E{\bigl (}x^{4}{\bigr )}\,-\,{\frac {(N-1)(N-3)}{N^{3}}}\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dbd6c48ac712c81ebf2fc7326ec98d5a1521b9)
![{\displaystyle ={\frac {N-1}{N^{3}}}\,\left\{(N-1)\,E{\bigl (}x^{4}{\bigr )}\,-\,(N-3)\,\left[E{\bigl (}x^{2}{\bigr )}\right]^{2}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd11a89fcf7fa5fc573a7790c8e7e3bdf2be5108)
