= Sampling uncertainty of an endpoint diffusivity estimate
{title2=$3n\widehat D/D\sim\chi^2_{3n}$}
Each Cartesian Brownian increment has <variance> $2Dt$. Summing their normalized squares gives a <chi-squared distribution> with $3n$ <degrees of freedom>. Hence $\widehat D$ is unbiased and has relative <standard deviation> $\sqrt{2/(3n)}$. Its inverse is not unbiased: for $3n>2$, $\mathbb E[1/\widehat D]=[3n/(3n-2)]/D$. Thus the direct inverse-diffusivity <radius> estimate has appreciable finite-sample bias when only a few trajectories are observed. Measurement noise or drift adds separate systematic error.
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