This question is a bit vague, but I was wondering how $L_2$ errors are distributed when the mean $L_2$ error is close to zero. For instance, if I have a point in $\mathbb{R}^2$ and a randomly, let's say uniformly randomly about mean distance of $\mu=5\times 10^{-3}$. If I call this random point $\vec{X}$ and the point that it's near $\vec{p}$, then the distance is $$ D = \sum_{i=1}^2 (p_i - X_i)^2 = \sum_{i=1}^2 \left(p_i^2 - 2p_i X_i + X_i^2\right) $$ I don't know how I can talk about how $D$ will be distributed from this. I read on Does the square of uniform distribution have density function? that $X_i$ should be distributed as $f_{X_i^2}(x) = \dfrac{5}{\sqrt{x}}$. The problem is I don't entirely understand the assertion made in the referenced question about the CDF of $X_i^2$. Regardless though, I don't know how I should combine $2p_i X_i$ with $X_i^2$. Furthermore, if I allowed for all points away from $p_i$ to be included in as a possible value for $X_i$, what would be a reasonable distribution to use for $X_i$ itself?
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"Uniformly randomly about a mean distance $\mu$" Could you be more specific? Is this meant to be uniformly over a circle of some radius centered on $p$? Somewhere else? And what's the radius? – eyeballfrog Mar 03 '23 at 16:16
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The tag distribution-theory is not about probability distributions. – md2perpe Mar 03 '23 at 17:51
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@eyeballfrog Yes, uniformly distributed points in a circle centered at $(p_1,p_2)$ up to a max distance away from $p$ of $0.01$. – David G. Mar 03 '23 at 18:42