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Say I sample data pairs {x,y} from a normal random variable with known mean and variance. What is the limiting distribution of |x-y|? [with what mean and variance?]

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If the samples are drawn independently from the same normally distributed random variable, then $X_1-X_2$ has a normal distribution with zero mean and twice the variance.

The absolute value $|X_1-X_2|$ therefore has a half-normal distribution

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