+ Y. Then the CDF for Z will be This integral is over the half-plane which lies under the line x+y = z. The key observation is that the function is radially symmetric. So we rotate the coordinate plane about the origin, choosing new coordinates such that the line x+y = z is described by the equation where is determined geometrically. Because of the radial symmetry, we have , and the CDF for Z is This is easy to integrate; we find that the CDF for Z is To determine the value , note that we rotated the plane so that the line x+y = z now runs vertically with x-intercept equal to c. So c is just the distance from the origin to the line x+y = z along the perpendicular bisector, which meets the line at its nearest point to the origin, in this case . So the distance is , and the CDF for Z is , i.e., Now, if a, b are any real constants (not both zero) then the probability that is found by the same integral as above, but with the bounding line . The same rotation method works, and in this more general case we find that the closest point on the line to the origin is located a (signed) distance away, so that The same argument in higher dimensions shows that if then Now we are essentially done, because So in general, if then Correlated random variables In the event that the variables X and Y are jointly normally distributed random variables, then X + Y is still normally distributed (see Multivariate normal distribution) and the mean is the sum of the means. However, the variances are not additive due to the correlation. Indeed, where ρ is the correlation. In particular, whenever ρ < 0, then the variance is less than the sum of the variances of X and Y. Extensions of this result can be made for more than two random variables, using the covariance matrix. Proof In this case (with X and Y having zero means), one needs to consider As above, one makes the substitution This integral is more complicated to simplify analytically, but can be done easily using a symbolic mathematics program. The probability distribution fZ(z) is given in this case by where If one considers instead Z = X − Y, then one obtains which also can