# Difference between revisions of "Documentation:Monte Carlo Equilibration"

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(→Error in slope of best-fitted line) |
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Denoting <math>\mathrm{Var}(y_i) = \sigma_{y}^2 </math>, we have: | Denoting <math>\mathrm{Var}(y_i) = \sigma_{y}^2 </math>, we have: | ||

− | <math> \Rightarrow \mathrm{Var}({\beta_1}) = \frac{\sigma_y^2}{\sum_i (x_i - \bar{x}_i)^2} </math> | + | <math> \Rightarrow \mathrm{Var}({\beta_1}) = \sigma_\beta^2 = \frac{\sigma_y^2}{\sum_i (x_i - \bar{x}_i)^2} </math> |

− | <math> \Rightarrow \mathrm{Var}({\beta_1}) = \frac{ 12 \, \sigma_y^2 }{ N(N^2 - 1) } </math> | + | <math> \Rightarrow \mathrm{Var}({\beta_1}) = \sigma_\beta^2 = \frac{ 12 \, \sigma_y^2 }{ N(N^2 - 1) } </math> |

## Revision as of 12:32, 9 September 2013

## Contents

# Monte Carlo equilibration

## Theory

We have a timeseries of N measurements obtained from a Monte Carlo simulation, i.e. .

Suppose (s.t. ) is the least-squares best fitted line, we attempt to minimize w.r.t. and .

, :

### Slope of best-fitted line

### Error in slope of best-fitted line

Denoting , we have: