The Normal Sum Theorem for a Linear Combination

  The normal sum theorem also works for a linear combination z=ax+by of normal variates x and y. Multiplication of a normal variate by a constant positive number produces another normal variate with proportional changes to the mean and standard deviation so a change in scale will reduce the problem to two cases z = x ± y but the minus sign doesn't affect the joint probability density.

Consequentially the probability density for z in both cases above will be the same after the integration over w irregardless of the sign of y and the variance of z will remain the sum of the two individual variances. For the linear combination var(z) = a² var(x) + b² var(y).