Does lim (t to infty) E(x-x hat) = 0 imply that there will be less disturbance over time?

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E(x)\, here denotes the expected value of x\,, where x\, is a random variable. \lim _{{t\to \infty }}E(x-{\hat  x})=0\, only implies that the mean estimation error will converge to zero over time. Disturbance, however, will affect the variance of the error, which is given by E(x-{\hat  x})^{2}\,. In CDS 110b, we will learn how to design observers that minimize this estimation variance if the disturbance can be modeled as Gaussian noise.