16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde ∑益x∑ 1, an unbiased estimate error=0 Ⅴar since the central statistics of the =k are those of the ng which are independent But note that the standard deviation of is o, so the standard deviation of k is 1 and that of The variance of -k is then Var(x) An easier way to remember this result is The addition of any new piece of data, no matter how large its variance, thus reduces the variance of x In the special case of equal quality data, o=on k the ordinary average, or arithmetic mean, of the data.16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 8 2 2 1 1 2 2 1 1 1 ˆ 1 1 N N k k k k k N N k k k k z x x x σ σ σ σ = = = = == = ∑ ∑ ∑ ∑ , an unbiased estimate error 0 = 2 1 2 2 1 Var Var( )ˆ 1 N k k k N k k z x σ σ = = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑ ∑ since the central statistics of the k z are those of the k n which are independent. But note that the standard deviation of k z is σ k , so the standard deviation of k k z σ is 1 and that of 2 k k z σ is 1 σ k . The variance of 2 k k z σ is then 2 1 σ k . 2 1 2 2 2 1 1 1 1 Var( )ˆ 1 1 N k k N N k k k k x σ σ σ = = = = = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑ ∑ ∑ An easier way to remember this result is 2 2 ˆ 1 1 1 N σ x k k= σ = ∑ The addition of any new piece of data, no matter how large its variance, thus reduces the variance of xˆ . In the special case of equal quality data, σ k n = σ 2 1 1 2 1 1 1 ˆ 1 1 N k N n k N k k n k z x z N σ σ = = = = = ∑ ∑ ∑ the ordinary average, or arithmetic mean, of the data