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Normal Difference Distribution

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Amazingly, the distribution of a difference of two normally distributed variates X and Y with means and variances (mu_x,sigma_x^2) and (mu_y,sigma_y^2), respectively, is given by

P_(X-Y)(u)=int_(-infty)^inftyint_(-infty)^infty(e^(-x^2/(2sigma_x^2)))/(sigma_xsqrt(2pi))(e^(-y^2/(2sigma_y^2)))/(sigma_ysqrt(2pi))delta((x-y)-u)dxdy
(1)
=(e^(-[u-(mu_x-mu_y)]^2/[2(sigma_x^2+sigma_y^2)]))/(sqrt(2pi(sigma_x^2+sigma_y^2))),
(2)

where delta(x) is a delta function, which is another normal distribution having mean

mu_(X-Y)=mu_x-mu_y
(3)

and variance

sigma_(X-Y)^2=sigma_x^2+sigma_y^2.
(4)

See also

Normal Distribution, Normal Ratio Distribution, Normal Sum Distribution

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Cite this as:

Weisstein, Eric W. "Normal Difference Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NormalDifferenceDistribution.html

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