Estimating variance
Estimating variance from a simulation ("on the fly")
where and
in the simulaton we keep track of the sum of x's and x^2's...
Or we can use E[((x - x_0) - x_1)]^2
Estimating variance from a simulation ("on the fly")
![{\displaystyle \sigma ^{2}=E[(x-{\bar {x}})^{2}]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/631833cab2f0c40a7ab32f54329c664f9235eba0)
![{\displaystyle =E[x^{2}]-2{\bar {x}}E[x]+{\bar {x}}^{2}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/413f2c6a94eb8e90ea74fcb9ee34d3433f0b384e)
![{\displaystyle =E[x^{2}]-2{\bar {x}}^{2}+{\bar {x}}^{2}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c02fb8b4a21ab0d881cc1746a2120629e6129aa9)
![{\displaystyle =E[x^{2}]-{\bar {x}}^{2}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0d3419a68f99504841c5fb107ab14c7a31ca308f)
where ![{\displaystyle E[x^{2}]={\frac {\sum _{i=1}^{N}x_{i}^{2}}{N}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/528cb318ce5abcd295e57458f19c81b34c363af1)
![{\displaystyle {\bar {x}}=E[x]={\frac {\sum _{i=1}^{N}x_{i}}{N}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/193a4e470452007fe77118d8d68c5d5f40bcc683)
in the simulaton we keep track of the sum of x's and x^2's...
Or we can use E[((x - x_0) - x_1)]^2